## Abstract

The irradiance of a partially coherent light propagated under the influence of multiple random effects is shown to be the convolution of the irradiance propagated in a vacuum with the system’s point spread function representing the random effects. This is true regardless of whether the propagation is far-field or not. We also show that the far-field irradiance of any laser system, regardless of complexity, can be expressed in terms of three basic parameters; laser power, field area, and a pupil factor. A general analytical formula for the far-field irradiance distribution for partially coherent laser sources of any complexity is derived. The formula includes multiple random effects including strong turbulence, random beam jitter, partial coherence, in addition to laser system pupil effects. An efficient matrix based numerical solution is also developed to verify the accuracy of the formula. Applications to the propagation of clipped Gaussian or flat-top beams with an obscuration, both as a single beam or an array of beams, are shown to give accurate results over the whole range of weak to strong turbulence as compared to numerical modeling.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## Corrections

Sami A. Shakir, Timothy T. Clark, Daniel S. Cargill, and Richard Carreras, "Far-field propagation of partially coherent laser light in random mediums: erratum," Opt. Express**26**, 21019-21019 (2018)

http://proxy.osapublishing.org/oe/abstract.cfm?uri=oe-26-16-21019

## 1. Introduction

The propagation of laser light from a laser source in turbid media, to a target or a receiver is an important element of most laser applications. Areas like communications, weaponry, ranging, remote sensing, laser radar, and medical applications [1–14], involve propagating a partially coherent laser beam through a medium that affects the laser beam before it reaches the intended target [1–9]. Diffraction is the primary physical effect that impacts the laser beam during propagation even if the propagation is in vacuum [1–4]. Hence, an accurate quantification of the impact of diffraction in the presence of other effects is critical for an accurate estimate of the laser irradiance delivered to a target or a receiver.

In recent years, researchers have been trying to simplify the propagation of light beams through random media [7]. Propagation problems involving random phase disturbances that might be rooted in the laser source optical chain itself or due to propagation in atmosphere are mathematically complex and difficult to model accurately [5,6]. This makes the application of numerical modeling essential at the cost of less transparency regarding the physics of the problem. In fact, analytical formulas that are exact or approximate are known for only a limited number of ideal cases, such as a point source, a plane wave and a perfectly Gaussian beam source [5,6]. Hence, the presence of a mathematically simple analytical formulas that capture the actual physical conditions and complexity of a problem, including the nature of the light used in addition to the nature of the propagation medium, is a useful addition. The work reported here attempts to achieve this goal for far-field propagation in the atmosphere including other effects such as platform jitter and inherent laser random effects that may result in partial coherence of the laser beam.

In this work, we will focus on proving some relations that are applicable to the propagation of partially coherent light in the far-field (FF-PC). The relations derived in this work are of general validity regardless of the shape of the laser beam or the strength of atmospheric turbulence. To achieve this goal, we will first review the formulas needed to prove the reported results in section 2. In section 3, we first establish the general result that the irradiance of a partially coherent laser beam propagating in a turbulent medium, can be viewed as the correlation of the irradiance in the absence of random effects with the Point Spread Function (PSF) of the random effects. Later in section 3, some simple and general formulas for propagated irradiance in the presence of random phase effects are derived. We also derive two matrix based numerical approaches that offer some advantages over Fast Fourier Transform (FFT) based methods. The validity and accuracy of the derived formulas are demonstrated by a comparison to numerically modeled results in section 4.

## 2. Basic field and irradiance equations

In this section we will review some basic results [5,6] needed to derive the main results of this work. In the analysis of light wave propagation in atmospheric turbulence, it has been customary to assume either weak turbulence or strong turbulence. In general, weak fluctuation theory is valid only when log-amplitude variance, ${\sigma}_{\chi}{}^{2}$, is less than about 0.2-0.5, otherwise the fluctuation is considered strong [5,6]. Several approaches have been proposed to deal with strong fluctuations. These include the diagram approach [15,16], the integral equation approach [17], the extended Huygens-Fresnel principle [18], and the parabolic or paraxial wave equation method [19]. Several works [5,6] have shown that for the second moment of the propagated field, these techniques are equivalent to each other under certain assumptions. For this reason, we will employ the extended Huygens-Fresnel approach because of its relative simplicity and employ parameters that are valid in both weak and strong fluctuations.

In a recent article, Gbur [14] reviewed the extended Huygens-Fresnel equation. The extended Huygens-Fresnel method treats the extended atmospheric turbulence between a source plane at lateral location z = 0 to an observation plane at z as a phase distortion of the spherical wave in the standard Huygens-Fresnel integral [1–6] such that the monochromatic, coherent source field $U({r}_{1},z=0)$ after propagating a distance z is given by,

The term $\psi ({\rho}_{1},r,z)$ represents the complex phase distortion imparted to the Huygens wavelet by atmospheric turbulence extended over the propagation distance z. The mean irradiance of the propagated field can be derived from the second moment of the field as [6,7]

If we assume that a diffuser is placed at the exit aperture of a coherent laser transmitter, we can model the field of the optical wave emerging from the diffuser by

where ${U}_{o}\left({\rho}_{1}\right)$ is the field entering the diffuser, ${U}^{\text{'}}{}_{o}\left({\rho}_{1}\right)$is the field exiting the diffuser, and $\varphi \left({\rho}_{1}\right)$is a random phase with zero average. In this setting we refer to ${U}_{o}\text{'}\left({\rho}_{1}\right)$as a partially coherent effective source and if we adopt the Gaussian Schell-model, the DOC or the MTF in this case is [6]

where ${\rho}_{PC}{}^{2}$is the effective variance of the diffuser correlation radius.## 3. Derived propagation equations

Using the basic field and irradiance equations of the previous section, we will derive some wave propagation results that are exact and general within the Fresnel paraxial approximation. Where the results are not exact, we will highlight that.

#### 3.1 System engineering model

The mean irradiance of a propagated field under the influence of atmospheric turbulence is the convolution of the irradiance in the observation plane without the atmosphere, with the overall Point Spread Function (PSF) of the extended atmosphere including all M- effects as,

where ${I}_{o}(r,z)$is the irradiance of the field propagated to the observation plane in the absence of effects other than diffraction, and ${P}_{M}(r,z)$is the PSF of all M random effects. The $\ast \ast $ symbol represents two-dimensional convolution. The effects’ PSF are related to their MTF’s or DOC’s through the Fourier Transform (F.T.) as,This is a general expression that is valid within the Fresnel paraxial approximation for all ranges, including the far-field. It is applicable to partially coherent beams including fully coherent beams. The similarity to the well-known incoherent imaging expression is striking. In fact, one may argue that the incoherent source case is a special case of Eq. (13).

#### 3.2 Far-field peak irradiance in vacuum

To show a very useful far-field result applicable to any laser source and of any geometry, we will first show that for any coherent source, the peak far-field irradiance for propagation under the influence of diffraction alone is [21]

where${P}_{L}$ is the laser source power; ${A}_{o}$is the field amplitude effective area, and ${F}_{o}$ is the pupil-factor which represents possible presence of obscurations in the exit pupil of the laser system. To show this expression, we assume a laser source with uniform phase that is focused at a plane a distance z away after passing through an exit pupil of the source system. The on-axis irradiance, ${I}_{PK}(z)={\left|U(0,z)\right|}^{2}$, follows Eq. (1) which leads to the equation,To accommodate general near-fields and geometries we introduce the notion of the field effective-area${A}_{o}$ in units of area, and the pupil factor ${F}_{o}$ defined as [21],

The field effective area and pupil factor are straightforward to derive for common realistic geometries [21]. Results for some of the more common systems, are summarized in Table 1.

The implication of Eq. (20) is that regardless of the shape and complexity of a laser field source, the far-field peak irradiance will be the same for the same wavelength, field area and output power. This is illustrated in Fig. (1) for two different beam shapes. One beam is a uniform or flat-top beam with a central obscuration and the second is a Gaussian beam with a central obscuration. They are assumed to have the same wavelength of 1 micron. The Flat-top beam has a diameter of 15 cm and the Gaussian beam a (1/e) field diameter of 10.6 cm. The obscuration radius is 3 cm for the flat-top beam and 2.17 cm for the Gaussian beam. The laser beam power before the obscuration is 1.0 W for both. The effective field area for both cases is 0.018 m^{2} and the pupil factor is 0.837. These source parameters result in an identical peak irradiance of 123.8 W/m^{2} for both cases when focused at a range of 10 km. The far-field irradiance profile for propagation in vacuum for a range of 10 km is calculated by a wave propagation program using the EMA algorithm described in section 3.4. While both have the same far-field peak irradiance, the impact of the source field shape shows up in the side lobes away from the optical axis. As we will show later, this side lobe difference essentially vanishes when strong turbulence or beam jitter is present. A similar washout effect takes place when the beam is partially incoherent.

#### 3.3 General far-field peak irradiance

The mean peak far-field irradiance of propagation under the influence of multiple random effects that are mutually independent is given by,

where S is the Strehl factor which satisfies the relation, $0\le S\le 1$.To prove Eq. (24), we start with Eq. (17) and set the observation point to the optical axis such that $r=0$. Hence, we have

Applying the mathematical mean-value theorem to Eq. (25) we obtain

In the Appendix we show that for a Gauss-Schell type partially coherent Gaussian beam of field source (1/e) radius ${w}_{o}$and net laser power *P _{L}*, which is propagated under the influence of independent random phase fluctuation effects of partial coherence, beam tilt jitter, and atmospheric turbulence, the irradiance of the beam propagated a distance z that may be different than the focal length ${f}_{L}$ of the focusing lens, is

We make the ansatz that since all beams follow Eq. (24), including Gaussian beams, then the Strehl function of Eq. (29) should also be valid for beams of any shape and complexity. From this ansatz and Eq. (27) applied to the far-field where $z={f}_{L}$, we get the general expression,

Equation (30) and Eq. (29) and Eq. (31) are the central results of this work and are essentially exact for Gaussian beam sources and expected to be very accurate for other laser geometries in the far-field. What these equations are implying is that for any laser system, if we know the source field area ${A}_{o}$, the system’s exit pupil factor ${F}_{o}$, and the laser power (before the exit pupil) ${P}_{L}$, then we can predict the far-field peak irradiance and irradiance distribution, for an arbitrary number of independent random effects. While we derived these results assuming strong turbulence, since the atmosphere’s MTF Eq. (9) is valid for all levels of turbulence [5,6], then Eq. (30) is valid under all levels of turbulence. Before we demonstrate the accuracy of these equations using numerical simulation, we develop an efficient numerical solution of the general wave equations of Eq. (1) and Eq. (3).

#### 3.4 Numerical model

The standard Huygens-Fresnel diffraction integral can be represented by an equivalent discrete matrix equation of the form [22],

where ${U}_{o}$is an (N_{o}x N

_{o}) matrix representing the complex source field,$U$is an (N

_{1}x N

_{1}) matrix representing the propagated complex field, and $H$is an (N

_{o}x N

_{1}) matrix representing the diffraction PSF as defined by Eq. (2). The “T” super script signifies the matrix transpose operation. The integer N

_{o}is the number of grid points along x and y axis, in the source plane, and N

_{1}is the number of grid points along the x-axis and y-axis, in the observation plane. Since we are multiplying matrices of dimension N x N rather than the usual N

^{2}x N

^{2}, the efficiency of this propagation approach compares competitively with Fast Fourier Transform (FFT)-based methods [22]. We call this matrix approach EMA (Efficient Matrix Algorithm) and a detailed discussion can be found in ref. (22). The efficient matrix formulation of Eq. (32) is possible because the PSF under the Fresnel approximation is separable in x and y.

The irradiance of the source field propagated a distance z in the presence of multiple random effects is given by Eq. (3). This involves four integrals and is generally challenging to program and solve numerically. However, a solution based on Eq. (17) offers a more efficient approach. Following the same procedure as above, one can show that the solution for the mean irradiance at any propagation distance z, (not just far-field), can be performed as,

where $D$is the $({N}_{o}x{N}_{o})$ matrix representing the DOC as defined in Eq. (7), and $J$is the $({N}_{o}\text{x}{N}_{1})$matrix defined by the function,_{o}x N

_{o}) matrix representing the modified source field as defined by Eq. (16). Note that in Eq. (33), the product $\left(Q.D\right)$is an element by element product and not a matrix product. The numerical implementation of Eq. (33) is similar to the EMA implementation [22] based on Eq. (32) and we call it Extended EMA, or XEMA. Both EMA and XEMA are straightforward to program using a matrix based numerical platform like Matlab.

## 4. Numerical verification of the FF-PC

To verify the accuracy of the general formula of Eq. (30) and Eq. (31) we use the XEMA approach to simulate various laser source configurations with realistic conditions including a Gaussian beam with obscurations and clipping and an array of flat-top beams with obscurations. In the following examples, we show propagation distances of up to 10 km for three cases of turbulence where the refractive index structure constant values are${C}_{n}{}^{2}=0\text{,}{\text{10}}^{-15}{m}^{-2/3},\text{and}5x{10}^{-15}{m}^{-2/3}$ . These values result in Rytov variances [6] corresponding to turbulence strength ranging from zero (i.e. vacuum) to 12, corresponding to very strong turbulence as the range is varied from 1km to 10 km. For constant refractive index structures, the Rytov variance is given by [6]; ${\sigma}_{R}{}^{2}=1.23{C}_{n}^{2}{k}^{7/6}{z}^{11/6}$, where k is the optical wave number and z is the propagation path length.

The first example shown in Fig. (2) which demonstrates the far-field peak irradiance equivalence of three different sources (1) a uniform or flat-top beam, (2) Clipped Gaussian beam with an obscuration, and (3) a flat-top with an obscuration. The fundamental parameters of the three systems; $\left\{{P}_{L},{F}_{o},{A}_{o}\right\}$, i.e. {power, field area, and pupil factor} are chosen such that their product is approximately 0.01. All three beams have a 1/e radius ${w}_{o}$ of 5 cm, and a wavelength of $1.0\mu m$ . The Gaussian beam is clipped at $1.8{w}_{o}$ and has an obscuration radius of $0.2{w}_{o}$ The flat-top beam with an obscuration, has an obscuration radius of $0.4{w}_{o}$ . The results shown in Fig. (2) represent the numerical computation using XEMA for the far-field peak irradiance as a function of propagation range. Turbulence was assumed to be constant at a relatively strong value of ${C}_{n}{}^{2}=5x{10}^{-15}{m}^{-2/3}$ . As expected, all three sources essentially have the same peak irradiance over the whole range.

The next example compares the prediction of Eq. (30) to numerical solution using Eq. (33). We assume a laser source with a Gaussian profile with a central obscuration and clipped at 1.8 times the beam 1/e field radius, ${w}_{o}$. The beam parameters are; a (1/e) radius ${w}_{o}$ of 5 cm, and a wavelength of $1.0\mu m$ and an obscuration radius of $0.1{w}_{o}$. The three turbulence scenarios correspond to a refractive index structure constant values of;${C}_{n}{}^{2}=0\text{,}{\text{10}}^{-15}{m}^{-2/3},\text{and}5x{10}^{-15}{m}^{-2/3}$. These values span cases of no turbulence to strong turbulence. In Fig. (3a), the peak far-field irradiance vs. propagation range for the three turbulence strengths is shown. The solid lines are the predictions of numerical code XEMA while the solid crosses correspond to the predictions of the analytical formula of Eq. (30). In Fig. (3b) we show the far-field transverse irradiance profile as a function of the x-axis for the three strengths of turbulence. Note that the predictions of analytical Eq. (30) (crosses) agree well with the numerical XEMA (solid lines) even though the turbulence strength spans the whole range from vacuum to very strong turbulence. In the whole range of turbulence strengths, the analytical formula agrees with the numerical wave optics based XEMA predictions to better than 3% for the peak irradiance predictions.

For laser arrays, the next example shown in Fig. (4) shows the peak far-field irradiance vs. propagation range for a close-packed array of four flat-top beams with an individual beam radius of ${w}_{o}=5\text{cm}$, obscuration radius $0.1{w}_{o}$, and a wavelength of $1.0\mu m$. The propagation distance ranges from 1.0 km to 10 km, corresponding to a Rytov variance ranging from 0 to 12. Again, we see a good agreement between the analytical prediction of Eq. (30) and numerical calculation of Eq. (33).

## 5. Conclusion

We have shown that the irradiance of a partially coherent light propagated under the influence of multiple random effects is the convolution of the irradiance propagated in absence of these effects, with the point spread function representing the random effects. This is generally true regardless of whether the propagation is far-field or not. We also show that the far-field irradiance of any laser system, regardless of complexity, can be expressed in terms of three basic parameters; laser power, field area, and a pupil factor. Sources having the same out-coupled power, pupil effect, and field area, will have the same peak irradiance and profile around the optical axis in the far-field. The far-field irradiance profile is smoothed out by the basically Gaussian shaped PSF of the composite random effects. The coherent properties of the source matter only in that they define the beam shape and size in the observation plane. For example, Gauss–Schell beams can have different initial sizes, focal distances, and coherence radii while maintaining the same spot size at a range z. The average irradiance distribution will be the same as for all these beams. This is even though turbulence is distributed along the path and interacts with the beam wave for all z. This also means that source beams having zeros in their field distribution, such as higher-order Gauss-Hermite laser modes with odd quantum numbers, end up with these zeros smoothed out in distant observation planes, especially when the strength of turbulence might be strong.

The mathematical complexity of problems involving propagation in the atmosphere, especially for problems involving realistic laser beam shapes, such as Gaussian beams with obscurations and wing clipping, or coherent arrays of laser beams, is replaced by a simple far-field formula (30). This far-field formula is applicable regardless of the shape and complexity of the laser source or the strength of the turbulence and could provide a convenient tool to accurately estimate the performance of complex laser systems under the presence of effects, such as partial coherence, platform jitter, and atmospheric turbulence.

We also took advantage of the separable nature of the PSF to cast the Huygens-Fresnel diffraction integral and the extended Huygens-Fresnel diffraction integral, in a convenient matrix form that compares favorably with FFT based propagation algorithms.

Atmospheric turbulence is assumed to be constant at ${C}_{n}{}^{2}=5x{10}^{-15}{m}^{-2/3}$.

## 6 Appendix: Gaussian beam propagation formula

To derive the mean irradiance of a propagated Gaussian field from the extended Huygens-Fresnel diffraction integral, it is customary to make a change of variables where $R={\scriptscriptstyle \frac{1}{2}}\left({r}_{1}+{r}_{2}\right)$ and $\rho =r{}_{1}-{r}_{2}$. In this case the propagated mean irradiance of Eq. (3) applied to a Gaussian source field, ${U}_{o}(r)=\left(2{P}_{L}/\pi {w}_{o}{}^{2}\right)\mathrm{exp}\left[-{r}^{2}/{w}_{o}{}^{2}\right]$ becomes [6],

## Funding

Air Force Research Laboratory (AFRL) at Kirtland Air Force Base.

## Acknowledgment

We also like to thank Thomas Dolash, Robin Ritter and Alan Andrews for helpful discussions.

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