Laser differential image-motion monitor for characterization of turbulence during free-space optical communication tests David M. Brown,* Juan C. Juarez, and Andrea M. Brown The Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Rd., Laurel, Maryland 20723, USA *Corresponding author: [email protected] Received 14 August 2013; accepted 8 October 2013; posted 24 October 2013 (Doc. ID 195766); published 25 November 2013
A laser differential image-motion monitor (DIMM) system was designed and constructed as part of a turbulence characterization suite during the DARPA free-space optical experimental network experiment (FOENEX) program. The developed link measurement system measures the atmospheric coherence length (r0 ), atmospheric scintillation, and power in the bucket for the 1550 nm band. DIMM measurements are made with two separate apertures coupled to a single InGaAs camera. The angle of arrival (AoA) for the wavefront at each aperture can be calculated based on focal spot movements imaged by the camera. By utilizing a single camera for the simultaneous measurement of the focal spots, the correlation of the variance in the AoA allows a straightforward computation of r0 as in traditional DIMM systems. Standard measurements of scintillation and power in the bucket are made with the same apertures by redirecting a percentage of the incoming signals to InGaAs detectors integrated with logarithmic amplifiers for high sensitivity and high dynamic range. By leveraging two, small apertures, the instrument forms a small size and weight configuration for mounting to actively tracking laser communication terminals for characterizing link performance. © 2013 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (010.7060) Turbulence; (010.7350) Wave-front sensing. http://dx.doi.org/10.1364/AO.52.008402
1. Introduction
The Johns Hopkins University Applied Physics Laboratory (JHU/APL) link measurement system (LIME) combines multiple techniques to characterize atmospheric turbulence during a free-space optical communication (FSOC) link and integrates them into a single system operating at the wavelength of the link. An inherent benefit of making measurements on the communications beam is that the approach supports measurements of a variety of link configurations by relying on the pointing and tracking system
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of the communications terminal. This expands the range of operations that can be supported because, in other approaches, separate sources and receivers are required (scintillometer, three-aperture scintillometer). When mobile links such as air-to-ground configurations are being tested, having separate instrumentation becomes impractical. Mobile link characterization is currently limited within the FSOC community. Some of the most common approaches are power-in-bucket (PIB) meters, angle of arrival (AoA) meters, and short-range (1–5 km) scintillometers. The atmospheric coherence length, r0 , is one of the most critical parameters affecting optical link performance, as it determines how much the beam spreads due to the size and
strength of turbulent eddies present along the path. Unfortunately, PIB meters do not provide a measurement of the coherence length of the atmosphere [1]. There are several well-established PIB techniques to calculate an inferred r0 along the path but no means of direct measurement of r0 [1,2]. While AoA meters and scintillometers measure the atmospheric structure constant, C2n , which can then be used to estimate r0 , their measurement approaches are limited to characterizing static paths. A differential image motion monitor (DIMM) is the most common tool for performing direct measurements of r0 and is one of the several instruments integrated into the JHU/APL LIME system. The measurement is performed by tracking the AoA variations of incoming light between spatially separate beam paths, yielding phase-distortion information. A multiaperture scintillometer is similar [3], and while this device does provide a measurement of path-integrated C2n , because it requires three or more apertures of varying sizes, it is impractical for physical insertion to the FSO communication beam. 2. Background
The developed system integrates several characterization tools (including a DIMM) into a compact sensor package operating at the wavelength of common FSOC systems [2] and provides a way to directly measure most the parameters required for evaluating link performance. All laser propagation programs seeking to detect or compensate for turbulence (high-energy laser systems, imaging applications, or FSOC) can leverage the developed approach for measuring turbulence along the path of interest. DIMM systems have been used for years to study the space to ground turbulence for astronomical applications. These systems are completely passive, using telescopes and starlight to measure the turbulence levels during astronomical observation [4,5]. DIMM techniques also can be employed for the characterization of turbulence horizontally through the atmosphere, assuming a stable remote source at range can be established [6]. For the JHU/APL DIMM system, this source is the optical communication beam as in the recent DARPA free-space optical experimental network experiment (FOENEX) program [2]. To gain a better understanding of the impact of turbulence on optical link performance, JHU/ APL engineers and scientists were tasked with measurement of path distortion during optical link testing. The JHU/APL DIMM sensor was designed by leveraging existing JHU/APL wave-optic-simulation tools to size the focal length and apertures of the receiver telescopes. The hardware was then constructed using commercial off-the-shelf components and tested during a 10 km link performance evaluation. By implementing twin apertures, the device provides aperture diverse scintillometer statistics, in addition to functioning as a DIMM, for measurement of atmospheric coherence length (r0 ). DIMM
instruments are typically realized by implementing a sub-aperture mask in the pupil plane of large astronomical telescopes [4,6]. This mask, when combined with shallow angle wedges in the optical path, can separate the light into two focal points [4]. Vibrations or other platform-related disturbances give rise to correlated motion of both focal points, while disturbances along the optical link path (turbulence) yield measurement of atmospheric coherence length. Differing from contemporary DIMM hardware, the JHU/APL DIMM instrument is designed to have a much smaller form factor, so that it can be interfaced with commercial optical communication terminals. The shrinking of the sensor form factor enables the instrument to provide vital link characteristics in a host of different system configurations and environments. For the FOENEX ground terminal, the small size of the developed instrument allowed it to be mounted close to the optical axis of the optical communications link, allowing measurements under a variety of turbulence conditions. If the JHU/APL DIMM were to mimic conventional DIMM systems, the large aperture would hinder measurements close to the optical communication link axis, and it would be limited to conditions with moderate to high turbulence, as the incoming beam would need to be spread enough to fill the optical communication terminal and the DIMM receiver. 3. Numerical Simulation of DIMM Sensor
Laser propagation and spread through a turbulent atmosphere can be simulated in a variety of ways, ranging from scaling laws [1] to high-fidelity wave optic solvers. Wave optic propagation techniques divide the atmosphere into many small propagation steps, inducing a small phase distortion for each longitudinal step for each time step [7,8]. The optical phase distortion for a given slab of atmosphere can be thought of as a 2D screen representing the integrated path differences experienced by a wavefront propagating normal to the slab for each x and y coordinate as illustrated in Fig. 1. The number of slabs (screens) needed to numerically propagate a laser beam through the flow field is often dictated by the strength of the distortion. The selection rules outlined by Knepp [9] and Martin and Flatté [10] are commonly used to determine the number of screens required for a propagation condition in addition to offering guidance on the selection of the transverse mesh cell sizes. The split step Fourier method (SSFM) is the engine behind optical wave propagation through the screens of atmosphere [7–11]. The SSFM is an efficient algorithm for the numerical solution of the nonlinear Schrödinger equation and principally relies on computing the solution in small steps along the propagation direction. Knowing that the nonlinear Schrödinger equation has linear and nonlinear parts, these terms are separated based on the assumption that only a small step in the propagation direction is taken. The linear part has an analytical solution in the time domain, so it can be solved directly for 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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Fig. 1. Split-step propagation through atmospheric turbulence.
this small step. The nonlinear part has an analytical solution in the frequency domain, so this term, which represents diffraction, is Fourier transformed and then commuted with the solution of the linear term. The inverse Fourier transform is then performed to return to the time domain, now having propagated a small step. The process is repeated until the complex wave field arrives at the destination or output screen. An example of a Gaussian laser beam after 10 km of free-space propagation without turbulence can be seen in the top right portion of Fig. 1. Turbulence is often expressed as pseudo-random phase screens spaced at an average of 100–200 m for most atmospheric conditions. The statistics of these phase screens are dictated by the Von Karman turbulence spectrum [7–13]. The approach used to generate these screens utilizes the Fourier transform for generation and is described in detail by Schmidt [11]. The turbulence-induced phase distortion presented in a Fourier-integral representation is Z ϕx; y
∞
−∞
Z
∞
−∞
Ψf x ; f y ei2πf x xf y y df x df y ;
(1)
where Ψf x ; f y is the spatial-frequency-domain representation of the phase. When implementing the simulation into MATLAB, the optical phase is written as a Fourier series giving ϕx; y
∞ ∞ X X
cn;m expi2πf xn x f ym y;
(2)
n−∞ m−∞
where cn;m are the Fourier series coefficients, and f xn and f ym are the discrete x and y directed spatial frequencies. Given that the Fourier series coefficients are complex with zero mean, equal variances, and zero cross-variances [13,14], they obey circular complex Gaussian statistics with zero mean and variance given by 8404
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hjcn;m j2 i Φϕ f xn ; f ym Δf xn Δf ym :
(3)
When implementing the above equation in MATLAB, the random function is employed to generate randn numbers with zero mean and unit variance with a Gaussian distribution. A few examples of phase screens can be found in the bottom portion of Fig. 1. The figure shows three phase screens for different levels of turbulence in units of radians, working from high to low turbulence, left to right. If a laser beam is propagated for 10 km through a homogeneous level of turbulence, namely C2n 10−15 m−2∕3, the complex distribution at the bottom right side in Fig. 1 is found. While only the real part of the field is shown in the figure (fluence), this distribution also contains the imaginary portion of the distribution, thereby describing the 2D phase of the beam at a 10 km range. These distributions are found for every time step of the simulation using different random number seeds. To facilitate the design of the laser DIMM sensor, a wave optic simulation was established for a nominal 10 km one-way path. The wave optic analysis begins with 500 independent realizations of the beam propagated to 10 km, representing statistically independent one-way propagations of the transmitted laser. For the transmitted wavelength of 1550 nm emanating from a 10.2 cm aperture, the DIMM was assumed to have two separate, co-aligned apertures with a separation that was parameterized to evaluate its performance. As the figure suggests, these atmospheric propagations are performed a priori for different levels of turbulence, assuming that the path is homogenous. After the atmospheric propagation step is performed, the complex beam pattern is multiplied by an aperture mask that describes the portion of the incoming beam that is collected by twin receiving apertures, separated by a parameterized distance. An example of the masked beam can be found in Fig. 2. (For
Fig. 2. Propagation internal to the DIMM receiver and calculation of atmospheric coherence length using simulated DIMM data.
simplicity, the focal plane image of only the left circular mask is shown in Fig. 2.) Following the application of the mask, the beam is independently propagated to the corresponding focal point for each of the two telescope receivers. The treatment of the problem where two separate propagation coordinate systems are employed is different from standard astronomical DIMM simulation. These systems use starlight as the remote source and therefore require much larger apertures. Typically these apertures are realized by placing a single mask in the pupil plane of a large aperture telescope. In this case, the twin receiving apertures share optical components along the telescope chain and therefore must be simultaneously propagated. Propagation to the focal plane of each of the matched telescopes uses 2D Fourier transforms. When the light propagates through the telescopes toward the focal plane, the ABCD matrix has values of A 0, B f , C f −1 , and D 1, assuming spherical optics [15]. To account for diffraction, a modified diffraction integral can be employed to simplify computations as compared to the more complex Huygens–Fresnel integral. The Fresnel diffraction integral can be simplified into a convolution to obtain the optical field in the focal plane: UAr2
1 iπβr2 2 e 2 Ur1 ⊗ eiπαr1 ; iλB
(4)
where α is defined as A∕λB and β as AC∕λ [16,17]. An example of the 2D image of the optical field in the focal plane for one of the DIMM arms can be found in Fig. 2, top middle. The optical field shown in the figure assumes a 1 μm resolution. Based on currently available InGaAs focal plane array receivers, this resolution is not possible; hence the focal plane optical field must be resampled. The top right focal plane image shows what would be observed in the focal plane based on commercially available cameras with 25 μm pixels. The same simulation is then performed
for the other arm of the DIMM sensor. The developed simulation then combines the focal plane images to a single matrix and saves them to memory, mimicking the real-world case of storing a series of frames from an IR camera. The centroid locations of the focal points from the left and right telescope arms are tracked over time to estimate the turbulence along the path. While not explicitly simulated, correlated motion of these focal points would be due to very large turbulence scales or, more importantly, DIMM platform vibration. Uncorrelated motion of these focal points is due to turbulence alone. The statistical variance in the focal point locations, σ 2XY , is used with the DIMM hardware parameters, including the operational wavelength, λ, the telescope focal length, f , the distance between the apertures, d, and the diameter of the apertures, D, to calculate the atmospheric coherence length for a given atmospheric path. Equation (5) shows this relationship [5,6]. 3 f 2 λ2 0.697d−1∕3 − 0.484D−1∕3 3∕5 m: r0 8 σ 2XY
(5)
If a homogenous horizontal path of turbulence is assumed, the atmospheric coherence length along with the path length, L, can be used to calculate the path-integrated atmospheric structure constant, C2n , as shown in Eq. (6) [1] −1
C2n 0.423k2 r05∕3 L m−2∕3 :
(6)
The statistical variance in the focal point locations, σ 2XY , can be described in terms of the mean square difference between the position vectors describing the focal points. For a single time step, the position vector for the centroid of the left or right focal spot in a common aperture can be denoted as the home location plus offset due to vibration, plus the offset 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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due to turbulence [Eq. (7)]. The difference vector due to turbulence can be found by first subtracting the appropriate offsets due to the focal point home locations. Next the vibration terms can be removed based on the assumption that they are equal, which is logical given that the left and right sides of the DIMM instrument share common mounting hardware. The variance of the resultant difference vector [Eq. (8)] is due to turbulence alone and is used to calculate the path-averaged atmospheric coherence length [Eq. (9)]. r⃗ lft r⃗ lft;home r⃗ lft;vib r⃗ lft;turb r⃗ rgt r⃗ rgt;home r⃗ rgt;vib r⃗ rgt;turb ;
(7)
r⃗ turb r⃗ lft;turb − r⃗ rgt;turb r⃗ lft − rlft;home − r⃗ lft;vib − ⃗rrgt − r⃗ rgt;home − r⃗ rgt;vib ; (8) σ 2XY h¯r2turb − r¯ 2turb i:
(9)
The focal point position vectors, or locations, are determined by performing independent centroid searches on both the left and right halves of the focal plane. The centroid algorithm used is a peak finder subroutine, which is executed after the simulated focal plane data has undergone a 2D spatial averaging filter with a five-pixel boxcar average. The result is
an x and y centroid location for both of the focal points in the focal plane of the DIMM sensor. 4. DIMM Hardware Design
The receiver propagation simulation was performed for several hardware configurations to optimize the combination of focal length, aperture sizes, and aperture separation to characterize turbulence with a target range of r0 1 to greater than 20 cm. Example results of this parameterized study can be found in Fig. 3. The cases shown in the figure encompass hardware configurations with large and small pixels and two different commercially available telescopes with different apertures and focal lengths. In both cases, the distance between the telescopes was fixed at 30 cm, which represents the upper-bound atmospheric coherence length that would be measureable with the designed DIMM sensor. A Fried parameter of 30 cm corresponds to benign turbulence conditions from a free-space laser propagation standpoint for the 10 km path under test. For each of the plots shown in Fig. 3, the x axis corresponds to the atmospheric coherence length or turbulence level entered into the wave propagation simulation. Side-by-side plots for each of the four system configurations in terms of r0 (left) and C2n (right) are shown. The entered turbulence value is used by the phase screen generation code within the atmospheric laser propagation module. The y axis corresponds to the path-averaged atmospheric coherence length or turbulence, which is calculated by performing the DIMM processing procedure
Fig. 3. Wave propagation simulation of proposed DIMM hardware using 500 independent realizations of turbulence. The boxed hardware configuration was constructed for testing. 8406
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described in the numerical simulation section herein. The simulation was performed for 500 time steps for each level of turbulence included in the study to ensure statistical relevance in the DIMM analysis of the simulated focal plane images. The left half of Fig. 3 shows the simulation results for two optical configurations using 1 μm pixels. The top pair of plots corresponds to a matched telescope pair that is 30 cm apart, each with a focal length of 1 m and an aperture diameter of 10 cm. The bottom pair represents a hardware configuration with the same 30 cm spacing between telescopes; however, the telescope focal lengths have been extended to 2 m, and the aperture diameters are reduced to 2.5 cm. In both hardware configurations, good performance across all turbulence levels is achieved, principally due to the small pixel size of the focal plane CCD imager. The right half of Fig. 3 shows additional simulation results with the pixel size increased to a more realistic 25 μm pixel pitch, keeping all other parameters the same. In the case of the 10 cm aperture and 1 m focal length, the simulated system struggles to detect any turbulence below 10−14, chiefly due to the lack of adequate resolution in the focal plane of the telescopes. The 25 μm pixel size is a systemdesign constraint due to commercial market limitations; therefore, the only way to improve the system performance is to extend the focal length of the telescope. Analogous to the left side of the figure, the bottom pair of plots represents a hardware configuration with 2 m focal lengths and aperture diameters of 2.5 cm. The increased optical lever arm of this configuration extends the simulated system performance down to turbulence levels of roughly 10−15, which is sufficient for the measurement application. The completed hardware design of the DIMM sensor is shown in Fig. 4. Different from astronomical DIMM systems, the JHU/APL laser DIMM sensor is able to use small collection apertures and still have adequate signal-to-noise because the remote source is a laser system, as opposed to starlight. All components in the DIMM system are standard BK-7 optics, AR coated for the 1550 nm band. The received light from the remote 1550 nm source is first collected by a pair of 5× Galilean beam reducers and then focused by a 400 mm plano convex spherical lens.
The effective focal length of the system is therefore 2 m. As the light traverses down the center of each of the two telescope arms, it slowly comes into focus after striking two 50∕50 beam splitters and two mirrors. The beams are combined to parallel optical paths prior to entering the optical receiver module using back-to-back 45 degree turning mirrors. The optical receiver consists of a light-tight enclosure with a 10 m FWHM bandpass filter in front of a Goodrich InGaAs CCD with 320 × 256 pixels. 5. DIMM Testing
The DIMM sensor was fielded in 2011 in support of the DARPA FOENEX program. The sensor was to provide path-integrated measurements of atmospheric coherence length, which would be used to assist with the validation of the FSOC link model. Over six days in May 2011, bidirectional links at 10 km were characterized during different periods of the day to evaluate link performance at different link ranges and under a variety of turbulence conditions. An image of the DIMM sensor deployed in the field is shown in Fig. 5. As shown on the right, the 1550 nm ground terminal with a 10.2 cm aperture, which served as the DIMM’s laser source, was located in Saratoga, California. The DIMM and an air terminal were located inside the AOptix Technologies facility in Campbell, California, 10 km away. The 1550 nm laser communication beam was directly measured with the DIMM sensor to characterize the effect of turbulence on laser communication. In addition to fielding the laser DIMM system, JHU/APL engineers co-located a short-wavelength infrared camera to image the beam distribution being captured by the FSOC terminal and the DIMM to gain additional insight into the turbulence level of the atmosphere. Other supporting instrumentation included a weather station and a near-infrared scintillometer that characterized local turbulence conditions by measuring C2n over a 175 m path outside the Campbell facility. Last, numerical predictions performed by Northrop Grumman based on the weather research and forecast (WRF) model were also used to predict r0 and C2n over the beam path based on actual weather conditions [18].
Fig. 4. Optical layout (left) and mechanical design (right) of the DIMM sensor. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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Fig. 5. DIMM sensor deployed in the field for turbulence characterization during the 10 km link testing (left) [2]. System diagram for the 10 km link testing for the FOENEX program (right).
Example processed DIMM data are shown in Fig. 6 for low turbulence (left) and high turbulence conditions (right). Top to bottom are single-frame images from the data-collection focal plane at two different times during the collection. The centroid was computed for both of the points in the focal plane and displayed as a white cross hair in the figure. The dashed blue circles in the figure are the bound-
ing case for the worst-case turbulence encountered, thereby serving as a reference point between the high and low turbulence case. In the low turbulence case, there is a clearly observable airy pattern and minimal movement in the location of the centroid when comparing the two frames top to bottom. In contrast, the high turbulence case shows clear deviation in the focal point location for the left and
Fig. 6. Example DIMM data collected during a recent field-testing exercise. Two individual frames captured at different times (top to bottom) during low turbulence (left) and high turbulence (right). 8408
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Fig. 7. Processed DIMM data compared to other measurements and model results [2].
right sides of the instrument. Because the motion of these focal point locations is not correlated, it is characterized as turbulence and folded into the measurement of atmospheric coherence length. The DIMM measurements from the evening of May 12, 2011, through the afternoon of May 13, 2011, are compared to turbulence estimates made using the WRF r0 predictions and by anchoring two commonly used C2n profile models to scintillometer measurements. An overlay of the four turbulence characterization approaches is presented in Fig. 7 [2]. The Hufnagel–Valley (HV) and the Hufnagel–Andrews–Phillips (HAP) C2n profile models were anchored to the ground-level scintillometer data collected during the test period to estimate r0 [19]. The height parameter for the modeled link was determined using level 2 digital terrain elevation data (30 m horizontal spacing). For both models, the wind parameter was fixed at a constant 21 m∕s, and the background turbulence parameter was set to one for the HAP model. The DIMM measurements and WRF estimates showed good agreement with both finding the smallest atmospheric coherence lengths to be about 1.4 cm during the peak of turbulence. During less turbulent periods, the DIMM and WRF approaches estimate r0 increased to nearly 10 cm. The HAP modeling approach was also in good agreement with the DIMM measurements and the WRF results, while the HV model applied to the same scintillometer data estimates much smaller r0 for all conditions. 6. Summary
To perform FSOC link diagnostics during the recent DARPA FOENEX program, JHU/APL engineers designed and built a link monitoring instrument. It included a custom-designed DIMM sensor operating at the wavelength of the FSOC link in addition to several tools used to measure link turbulence such as power-in-the-bucket meters. Wave propagation software developed for high-energy laser applications enabled JHU/APL to simulate the DIMM prior to construction and testing of the instrument. These
models were critical for evaluating the performance of the proposed hardware and were used to optimize the system design for measurement of the turbulence strength during FSOC link testing. The final hardware was constructed and fielded for a 10 km test alongside a ground-level scintillometer, measuring local turbulence. The local scintillometer data was fed into turbulence profile models and compared to the analyzed DIMM data. Additionally, WRF modeling was performed to offer an additional comparison metric. The measurement of atmospheric coherence length obtained using the DIMM instrumentation was consistent with the WRF and HAP scintillometer results. References 1. L. C. Andrews and R. L. Phillip, Laser Beam Propagation through Random Media (SPIE, 2005). 2. J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012). 3. L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012). 4. A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002). 5. D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988). 6. M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001). 7. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961). 8. R. Fante, “Electromagnetic beam propagation in Turbulent Media,” Proc. IEEE 63, 1669–1692 (1975). 9. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983). 10. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988). 11. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010). 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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12. A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008). 13. B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997). 14. W. A. Coles, J. P. Filice, R. G. Frehlich, and M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089–2101 (1995). 15. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
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16. A. J. Lambert and D. Fraser, “Linear systems approach to simulating optical diffraction,” Appl. Opt. 37, 7933–7939 (1998). 17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968). 18. B. Felton and R. Alliss, “Improved climatological characterization of optical turbulence for free-space optical communications,” Proc. SPIE 8162, 816204 (2011). 19. L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
A laser differential image-motion monitor (DIMM) system was designed and constructed as part of a turbulence characterization suite during the DARPA free-space optical experimental network experiment (FOENEX) program. The developed link measurement system measures the atmospheric coherence length (r0 ), atmospheric scintillation, and power in the bucket for the 1550 nm band. DIMM measurements are made with two separate apertures coupled to a single InGaAs camera. The angle of arrival (AoA) for the wavefront at each aperture can be calculated based on focal spot movements imaged by the camera. By utilizing a single camera for the simultaneous measurement of the focal spots, the correlation of the variance in the AoA allows a straightforward computation of r0 as in traditional DIMM systems. Standard measurements of scintillation and power in the bucket are made with the same apertures by redirecting a percentage of the incoming signals to InGaAs detectors integrated with logarithmic amplifiers for high sensitivity and high dynamic range. By leveraging two, small apertures, the instrument forms a small size and weight configuration for mounting to actively tracking laser communication terminals for characterizing link performance. © 2013 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (010.7060) Turbulence; (010.7350) Wave-front sensing. http://dx.doi.org/10.1364/AO.52.008402
1. Introduction
The Johns Hopkins University Applied Physics Laboratory (JHU/APL) link measurement system (LIME) combines multiple techniques to characterize atmospheric turbulence during a free-space optical communication (FSOC) link and integrates them into a single system operating at the wavelength of the link. An inherent benefit of making measurements on the communications beam is that the approach supports measurements of a variety of link configurations by relying on the pointing and tracking system
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of the communications terminal. This expands the range of operations that can be supported because, in other approaches, separate sources and receivers are required (scintillometer, three-aperture scintillometer). When mobile links such as air-to-ground configurations are being tested, having separate instrumentation becomes impractical. Mobile link characterization is currently limited within the FSOC community. Some of the most common approaches are power-in-bucket (PIB) meters, angle of arrival (AoA) meters, and short-range (1–5 km) scintillometers. The atmospheric coherence length, r0 , is one of the most critical parameters affecting optical link performance, as it determines how much the beam spreads due to the size and
strength of turbulent eddies present along the path. Unfortunately, PIB meters do not provide a measurement of the coherence length of the atmosphere [1]. There are several well-established PIB techniques to calculate an inferred r0 along the path but no means of direct measurement of r0 [1,2]. While AoA meters and scintillometers measure the atmospheric structure constant, C2n , which can then be used to estimate r0 , their measurement approaches are limited to characterizing static paths. A differential image motion monitor (DIMM) is the most common tool for performing direct measurements of r0 and is one of the several instruments integrated into the JHU/APL LIME system. The measurement is performed by tracking the AoA variations of incoming light between spatially separate beam paths, yielding phase-distortion information. A multiaperture scintillometer is similar [3], and while this device does provide a measurement of path-integrated C2n , because it requires three or more apertures of varying sizes, it is impractical for physical insertion to the FSO communication beam. 2. Background
The developed system integrates several characterization tools (including a DIMM) into a compact sensor package operating at the wavelength of common FSOC systems [2] and provides a way to directly measure most the parameters required for evaluating link performance. All laser propagation programs seeking to detect or compensate for turbulence (high-energy laser systems, imaging applications, or FSOC) can leverage the developed approach for measuring turbulence along the path of interest. DIMM systems have been used for years to study the space to ground turbulence for astronomical applications. These systems are completely passive, using telescopes and starlight to measure the turbulence levels during astronomical observation [4,5]. DIMM techniques also can be employed for the characterization of turbulence horizontally through the atmosphere, assuming a stable remote source at range can be established [6]. For the JHU/APL DIMM system, this source is the optical communication beam as in the recent DARPA free-space optical experimental network experiment (FOENEX) program [2]. To gain a better understanding of the impact of turbulence on optical link performance, JHU/ APL engineers and scientists were tasked with measurement of path distortion during optical link testing. The JHU/APL DIMM sensor was designed by leveraging existing JHU/APL wave-optic-simulation tools to size the focal length and apertures of the receiver telescopes. The hardware was then constructed using commercial off-the-shelf components and tested during a 10 km link performance evaluation. By implementing twin apertures, the device provides aperture diverse scintillometer statistics, in addition to functioning as a DIMM, for measurement of atmospheric coherence length (r0 ). DIMM
instruments are typically realized by implementing a sub-aperture mask in the pupil plane of large astronomical telescopes [4,6]. This mask, when combined with shallow angle wedges in the optical path, can separate the light into two focal points [4]. Vibrations or other platform-related disturbances give rise to correlated motion of both focal points, while disturbances along the optical link path (turbulence) yield measurement of atmospheric coherence length. Differing from contemporary DIMM hardware, the JHU/APL DIMM instrument is designed to have a much smaller form factor, so that it can be interfaced with commercial optical communication terminals. The shrinking of the sensor form factor enables the instrument to provide vital link characteristics in a host of different system configurations and environments. For the FOENEX ground terminal, the small size of the developed instrument allowed it to be mounted close to the optical axis of the optical communications link, allowing measurements under a variety of turbulence conditions. If the JHU/APL DIMM were to mimic conventional DIMM systems, the large aperture would hinder measurements close to the optical communication link axis, and it would be limited to conditions with moderate to high turbulence, as the incoming beam would need to be spread enough to fill the optical communication terminal and the DIMM receiver. 3. Numerical Simulation of DIMM Sensor
Laser propagation and spread through a turbulent atmosphere can be simulated in a variety of ways, ranging from scaling laws [1] to high-fidelity wave optic solvers. Wave optic propagation techniques divide the atmosphere into many small propagation steps, inducing a small phase distortion for each longitudinal step for each time step [7,8]. The optical phase distortion for a given slab of atmosphere can be thought of as a 2D screen representing the integrated path differences experienced by a wavefront propagating normal to the slab for each x and y coordinate as illustrated in Fig. 1. The number of slabs (screens) needed to numerically propagate a laser beam through the flow field is often dictated by the strength of the distortion. The selection rules outlined by Knepp [9] and Martin and Flatté [10] are commonly used to determine the number of screens required for a propagation condition in addition to offering guidance on the selection of the transverse mesh cell sizes. The split step Fourier method (SSFM) is the engine behind optical wave propagation through the screens of atmosphere [7–11]. The SSFM is an efficient algorithm for the numerical solution of the nonlinear Schrödinger equation and principally relies on computing the solution in small steps along the propagation direction. Knowing that the nonlinear Schrödinger equation has linear and nonlinear parts, these terms are separated based on the assumption that only a small step in the propagation direction is taken. The linear part has an analytical solution in the time domain, so it can be solved directly for 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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Fig. 1. Split-step propagation through atmospheric turbulence.
this small step. The nonlinear part has an analytical solution in the frequency domain, so this term, which represents diffraction, is Fourier transformed and then commuted with the solution of the linear term. The inverse Fourier transform is then performed to return to the time domain, now having propagated a small step. The process is repeated until the complex wave field arrives at the destination or output screen. An example of a Gaussian laser beam after 10 km of free-space propagation without turbulence can be seen in the top right portion of Fig. 1. Turbulence is often expressed as pseudo-random phase screens spaced at an average of 100–200 m for most atmospheric conditions. The statistics of these phase screens are dictated by the Von Karman turbulence spectrum [7–13]. The approach used to generate these screens utilizes the Fourier transform for generation and is described in detail by Schmidt [11]. The turbulence-induced phase distortion presented in a Fourier-integral representation is Z ϕx; y
∞
−∞
Z
∞
−∞
Ψf x ; f y ei2πf x xf y y df x df y ;
(1)
where Ψf x ; f y is the spatial-frequency-domain representation of the phase. When implementing the simulation into MATLAB, the optical phase is written as a Fourier series giving ϕx; y
∞ ∞ X X
cn;m expi2πf xn x f ym y;
(2)
n−∞ m−∞
where cn;m are the Fourier series coefficients, and f xn and f ym are the discrete x and y directed spatial frequencies. Given that the Fourier series coefficients are complex with zero mean, equal variances, and zero cross-variances [13,14], they obey circular complex Gaussian statistics with zero mean and variance given by 8404
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hjcn;m j2 i Φϕ f xn ; f ym Δf xn Δf ym :
(3)
When implementing the above equation in MATLAB, the random function is employed to generate randn numbers with zero mean and unit variance with a Gaussian distribution. A few examples of phase screens can be found in the bottom portion of Fig. 1. The figure shows three phase screens for different levels of turbulence in units of radians, working from high to low turbulence, left to right. If a laser beam is propagated for 10 km through a homogeneous level of turbulence, namely C2n 10−15 m−2∕3, the complex distribution at the bottom right side in Fig. 1 is found. While only the real part of the field is shown in the figure (fluence), this distribution also contains the imaginary portion of the distribution, thereby describing the 2D phase of the beam at a 10 km range. These distributions are found for every time step of the simulation using different random number seeds. To facilitate the design of the laser DIMM sensor, a wave optic simulation was established for a nominal 10 km one-way path. The wave optic analysis begins with 500 independent realizations of the beam propagated to 10 km, representing statistically independent one-way propagations of the transmitted laser. For the transmitted wavelength of 1550 nm emanating from a 10.2 cm aperture, the DIMM was assumed to have two separate, co-aligned apertures with a separation that was parameterized to evaluate its performance. As the figure suggests, these atmospheric propagations are performed a priori for different levels of turbulence, assuming that the path is homogenous. After the atmospheric propagation step is performed, the complex beam pattern is multiplied by an aperture mask that describes the portion of the incoming beam that is collected by twin receiving apertures, separated by a parameterized distance. An example of the masked beam can be found in Fig. 2. (For
Fig. 2. Propagation internal to the DIMM receiver and calculation of atmospheric coherence length using simulated DIMM data.
simplicity, the focal plane image of only the left circular mask is shown in Fig. 2.) Following the application of the mask, the beam is independently propagated to the corresponding focal point for each of the two telescope receivers. The treatment of the problem where two separate propagation coordinate systems are employed is different from standard astronomical DIMM simulation. These systems use starlight as the remote source and therefore require much larger apertures. Typically these apertures are realized by placing a single mask in the pupil plane of a large aperture telescope. In this case, the twin receiving apertures share optical components along the telescope chain and therefore must be simultaneously propagated. Propagation to the focal plane of each of the matched telescopes uses 2D Fourier transforms. When the light propagates through the telescopes toward the focal plane, the ABCD matrix has values of A 0, B f , C f −1 , and D 1, assuming spherical optics [15]. To account for diffraction, a modified diffraction integral can be employed to simplify computations as compared to the more complex Huygens–Fresnel integral. The Fresnel diffraction integral can be simplified into a convolution to obtain the optical field in the focal plane: UAr2
1 iπβr2 2 e 2 Ur1 ⊗ eiπαr1 ; iλB
(4)
where α is defined as A∕λB and β as AC∕λ [16,17]. An example of the 2D image of the optical field in the focal plane for one of the DIMM arms can be found in Fig. 2, top middle. The optical field shown in the figure assumes a 1 μm resolution. Based on currently available InGaAs focal plane array receivers, this resolution is not possible; hence the focal plane optical field must be resampled. The top right focal plane image shows what would be observed in the focal plane based on commercially available cameras with 25 μm pixels. The same simulation is then performed
for the other arm of the DIMM sensor. The developed simulation then combines the focal plane images to a single matrix and saves them to memory, mimicking the real-world case of storing a series of frames from an IR camera. The centroid locations of the focal points from the left and right telescope arms are tracked over time to estimate the turbulence along the path. While not explicitly simulated, correlated motion of these focal points would be due to very large turbulence scales or, more importantly, DIMM platform vibration. Uncorrelated motion of these focal points is due to turbulence alone. The statistical variance in the focal point locations, σ 2XY , is used with the DIMM hardware parameters, including the operational wavelength, λ, the telescope focal length, f , the distance between the apertures, d, and the diameter of the apertures, D, to calculate the atmospheric coherence length for a given atmospheric path. Equation (5) shows this relationship [5,6]. 3 f 2 λ2 0.697d−1∕3 − 0.484D−1∕3 3∕5 m: r0 8 σ 2XY
(5)
If a homogenous horizontal path of turbulence is assumed, the atmospheric coherence length along with the path length, L, can be used to calculate the path-integrated atmospheric structure constant, C2n , as shown in Eq. (6) [1] −1
C2n 0.423k2 r05∕3 L m−2∕3 :
(6)
The statistical variance in the focal point locations, σ 2XY , can be described in terms of the mean square difference between the position vectors describing the focal points. For a single time step, the position vector for the centroid of the left or right focal spot in a common aperture can be denoted as the home location plus offset due to vibration, plus the offset 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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due to turbulence [Eq. (7)]. The difference vector due to turbulence can be found by first subtracting the appropriate offsets due to the focal point home locations. Next the vibration terms can be removed based on the assumption that they are equal, which is logical given that the left and right sides of the DIMM instrument share common mounting hardware. The variance of the resultant difference vector [Eq. (8)] is due to turbulence alone and is used to calculate the path-averaged atmospheric coherence length [Eq. (9)]. r⃗ lft r⃗ lft;home r⃗ lft;vib r⃗ lft;turb r⃗ rgt r⃗ rgt;home r⃗ rgt;vib r⃗ rgt;turb ;
(7)
r⃗ turb r⃗ lft;turb − r⃗ rgt;turb r⃗ lft − rlft;home − r⃗ lft;vib − ⃗rrgt − r⃗ rgt;home − r⃗ rgt;vib ; (8) σ 2XY h¯r2turb − r¯ 2turb i:
(9)
The focal point position vectors, or locations, are determined by performing independent centroid searches on both the left and right halves of the focal plane. The centroid algorithm used is a peak finder subroutine, which is executed after the simulated focal plane data has undergone a 2D spatial averaging filter with a five-pixel boxcar average. The result is
an x and y centroid location for both of the focal points in the focal plane of the DIMM sensor. 4. DIMM Hardware Design
The receiver propagation simulation was performed for several hardware configurations to optimize the combination of focal length, aperture sizes, and aperture separation to characterize turbulence with a target range of r0 1 to greater than 20 cm. Example results of this parameterized study can be found in Fig. 3. The cases shown in the figure encompass hardware configurations with large and small pixels and two different commercially available telescopes with different apertures and focal lengths. In both cases, the distance between the telescopes was fixed at 30 cm, which represents the upper-bound atmospheric coherence length that would be measureable with the designed DIMM sensor. A Fried parameter of 30 cm corresponds to benign turbulence conditions from a free-space laser propagation standpoint for the 10 km path under test. For each of the plots shown in Fig. 3, the x axis corresponds to the atmospheric coherence length or turbulence level entered into the wave propagation simulation. Side-by-side plots for each of the four system configurations in terms of r0 (left) and C2n (right) are shown. The entered turbulence value is used by the phase screen generation code within the atmospheric laser propagation module. The y axis corresponds to the path-averaged atmospheric coherence length or turbulence, which is calculated by performing the DIMM processing procedure
Fig. 3. Wave propagation simulation of proposed DIMM hardware using 500 independent realizations of turbulence. The boxed hardware configuration was constructed for testing. 8406
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described in the numerical simulation section herein. The simulation was performed for 500 time steps for each level of turbulence included in the study to ensure statistical relevance in the DIMM analysis of the simulated focal plane images. The left half of Fig. 3 shows the simulation results for two optical configurations using 1 μm pixels. The top pair of plots corresponds to a matched telescope pair that is 30 cm apart, each with a focal length of 1 m and an aperture diameter of 10 cm. The bottom pair represents a hardware configuration with the same 30 cm spacing between telescopes; however, the telescope focal lengths have been extended to 2 m, and the aperture diameters are reduced to 2.5 cm. In both hardware configurations, good performance across all turbulence levels is achieved, principally due to the small pixel size of the focal plane CCD imager. The right half of Fig. 3 shows additional simulation results with the pixel size increased to a more realistic 25 μm pixel pitch, keeping all other parameters the same. In the case of the 10 cm aperture and 1 m focal length, the simulated system struggles to detect any turbulence below 10−14, chiefly due to the lack of adequate resolution in the focal plane of the telescopes. The 25 μm pixel size is a systemdesign constraint due to commercial market limitations; therefore, the only way to improve the system performance is to extend the focal length of the telescope. Analogous to the left side of the figure, the bottom pair of plots represents a hardware configuration with 2 m focal lengths and aperture diameters of 2.5 cm. The increased optical lever arm of this configuration extends the simulated system performance down to turbulence levels of roughly 10−15, which is sufficient for the measurement application. The completed hardware design of the DIMM sensor is shown in Fig. 4. Different from astronomical DIMM systems, the JHU/APL laser DIMM sensor is able to use small collection apertures and still have adequate signal-to-noise because the remote source is a laser system, as opposed to starlight. All components in the DIMM system are standard BK-7 optics, AR coated for the 1550 nm band. The received light from the remote 1550 nm source is first collected by a pair of 5× Galilean beam reducers and then focused by a 400 mm plano convex spherical lens.
The effective focal length of the system is therefore 2 m. As the light traverses down the center of each of the two telescope arms, it slowly comes into focus after striking two 50∕50 beam splitters and two mirrors. The beams are combined to parallel optical paths prior to entering the optical receiver module using back-to-back 45 degree turning mirrors. The optical receiver consists of a light-tight enclosure with a 10 m FWHM bandpass filter in front of a Goodrich InGaAs CCD with 320 × 256 pixels. 5. DIMM Testing
The DIMM sensor was fielded in 2011 in support of the DARPA FOENEX program. The sensor was to provide path-integrated measurements of atmospheric coherence length, which would be used to assist with the validation of the FSOC link model. Over six days in May 2011, bidirectional links at 10 km were characterized during different periods of the day to evaluate link performance at different link ranges and under a variety of turbulence conditions. An image of the DIMM sensor deployed in the field is shown in Fig. 5. As shown on the right, the 1550 nm ground terminal with a 10.2 cm aperture, which served as the DIMM’s laser source, was located in Saratoga, California. The DIMM and an air terminal were located inside the AOptix Technologies facility in Campbell, California, 10 km away. The 1550 nm laser communication beam was directly measured with the DIMM sensor to characterize the effect of turbulence on laser communication. In addition to fielding the laser DIMM system, JHU/APL engineers co-located a short-wavelength infrared camera to image the beam distribution being captured by the FSOC terminal and the DIMM to gain additional insight into the turbulence level of the atmosphere. Other supporting instrumentation included a weather station and a near-infrared scintillometer that characterized local turbulence conditions by measuring C2n over a 175 m path outside the Campbell facility. Last, numerical predictions performed by Northrop Grumman based on the weather research and forecast (WRF) model were also used to predict r0 and C2n over the beam path based on actual weather conditions [18].
Fig. 4. Optical layout (left) and mechanical design (right) of the DIMM sensor. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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Fig. 5. DIMM sensor deployed in the field for turbulence characterization during the 10 km link testing (left) [2]. System diagram for the 10 km link testing for the FOENEX program (right).
Example processed DIMM data are shown in Fig. 6 for low turbulence (left) and high turbulence conditions (right). Top to bottom are single-frame images from the data-collection focal plane at two different times during the collection. The centroid was computed for both of the points in the focal plane and displayed as a white cross hair in the figure. The dashed blue circles in the figure are the bound-
ing case for the worst-case turbulence encountered, thereby serving as a reference point between the high and low turbulence case. In the low turbulence case, there is a clearly observable airy pattern and minimal movement in the location of the centroid when comparing the two frames top to bottom. In contrast, the high turbulence case shows clear deviation in the focal point location for the left and
Fig. 6. Example DIMM data collected during a recent field-testing exercise. Two individual frames captured at different times (top to bottom) during low turbulence (left) and high turbulence (right). 8408
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Fig. 7. Processed DIMM data compared to other measurements and model results [2].
right sides of the instrument. Because the motion of these focal point locations is not correlated, it is characterized as turbulence and folded into the measurement of atmospheric coherence length. The DIMM measurements from the evening of May 12, 2011, through the afternoon of May 13, 2011, are compared to turbulence estimates made using the WRF r0 predictions and by anchoring two commonly used C2n profile models to scintillometer measurements. An overlay of the four turbulence characterization approaches is presented in Fig. 7 [2]. The Hufnagel–Valley (HV) and the Hufnagel–Andrews–Phillips (HAP) C2n profile models were anchored to the ground-level scintillometer data collected during the test period to estimate r0 [19]. The height parameter for the modeled link was determined using level 2 digital terrain elevation data (30 m horizontal spacing). For both models, the wind parameter was fixed at a constant 21 m∕s, and the background turbulence parameter was set to one for the HAP model. The DIMM measurements and WRF estimates showed good agreement with both finding the smallest atmospheric coherence lengths to be about 1.4 cm during the peak of turbulence. During less turbulent periods, the DIMM and WRF approaches estimate r0 increased to nearly 10 cm. The HAP modeling approach was also in good agreement with the DIMM measurements and the WRF results, while the HV model applied to the same scintillometer data estimates much smaller r0 for all conditions. 6. Summary
To perform FSOC link diagnostics during the recent DARPA FOENEX program, JHU/APL engineers designed and built a link monitoring instrument. It included a custom-designed DIMM sensor operating at the wavelength of the FSOC link in addition to several tools used to measure link turbulence such as power-in-the-bucket meters. Wave propagation software developed for high-energy laser applications enabled JHU/APL to simulate the DIMM prior to construction and testing of the instrument. These
models were critical for evaluating the performance of the proposed hardware and were used to optimize the system design for measurement of the turbulence strength during FSOC link testing. The final hardware was constructed and fielded for a 10 km test alongside a ground-level scintillometer, measuring local turbulence. The local scintillometer data was fed into turbulence profile models and compared to the analyzed DIMM data. Additionally, WRF modeling was performed to offer an additional comparison metric. The measurement of atmospheric coherence length obtained using the DIMM instrumentation was consistent with the WRF and HAP scintillometer results. References 1. L. C. Andrews and R. L. Phillip, Laser Beam Propagation through Random Media (SPIE, 2005). 2. J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012). 3. L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012). 4. A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002). 5. D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988). 6. M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001). 7. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961). 8. R. Fante, “Electromagnetic beam propagation in Turbulent Media,” Proc. IEEE 63, 1669–1692 (1975). 9. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983). 10. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988). 11. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010). 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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12. A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008). 13. B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997). 14. W. A. Coles, J. P. Filice, R. G. Frehlich, and M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089–2101 (1995). 15. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
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16. A. J. Lambert and D. Fraser, “Linear systems approach to simulating optical diffraction,” Appl. Opt. 37, 7933–7939 (1998). 17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968). 18. B. Felton and R. Alliss, “Improved climatological characterization of optical turbulence for free-space optical communications,” Proc. SPIE 8162, 816204 (2011). 19. L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
