The propagation effects of spatially pseudo-partially coherent Gaussian Schell-model beams in atmosphere are investigated numerically. The characteristics of beam spreading, beam wandering and intensity scintillation are analyzed respectively. It is found that the degradation of degree of source coherence may cause reductions of relative beam spreading and scintillation index, which indicates that partially coherent beams are more resistant to atmospheric turbulence than fully coherent beams. And beam wandering is not much sensitive to the change of source coherence. However, a partially coherent beam have a larger spreading than the fully coherent beam both in free space and in atmospheric turbulence. The influences of changing frequency of random phase screen which models the source coherence on the final intensity pattern are also discussed.
©2009 Optical Society of America
Research on laser beam propagating properties through atmospheric turbulence is very important for the analysis and performance prediction in many applications of laser engineering, such as remote sensing, tracking and free-space optical communications. High directionality and high coherence are the outstanding advantages of laser used in those applications. However, a fully coherent beam becomes partially coherent as it propagates in atmospheric turbulence, especially in strong turbulence. So knowledge of how a partially coherent beam propagates though turbulence is also vital for understanding completely laser propagating properties. On the other hand, fully coherent laser beams are very sensitive to atmospheric turbulence, as a consequence, turbulence effect becomes a main restricted factor in many laser engineering applications, such as the effects of beam spreading, wandering and intensity scintillation [1–6]. Recently, it was suggested theoretically that partially coherent beams (PCBs) may be less susceptible to turbulence than fully coherent beams (CBs) in many publications [7–12]. Thus the use of PCBs can be a possible way to improve the performance of laser engineering.
So far, most studies were concentrated on the Gaussian Schell-mode (GSM) beams with partial spatial coherence. The partial coherence is always modeled by a random phase screen which changes randomly with a Gaussian spatial coherence function . However, the GSM beam is an analytical tractable model and it is very difficult to create a fully-developed GSM beam in practice because of the limited changing frequency of the random phase screen. Nevertheless, changes in atmospheric turbulence are relatively slow compared to lasercom bit rates. It may not be necessary to create fully-developed partially coherent beams to improve the performance of laser engineering. Thus, the concept of “pseudo-partially coherent beam (PPCB)” was proposed by David Voelz and Kevin Fitzhenry to meet the need of practice [14, 15], which involved that a PPCB can be created by sending a CB through a random phase screen that changes only a few times over a bit duration, i.e. during the time that distortion of atmospheric turbulence has changed once this random phase screen has changed for a few times.
In spite of substantial theoretical research on the propagation of fully-developed GSM beams, the propagation of PPCBs does not seem to have been studied in detail. And it was pointed out that pseudo-partially coherent GSM beams may not follow the GSM theory and are better examined through wave optics numerical simulation. The main objective of this paper is to investigate numerically the propagation effects of spatially pseudo-partially coherent GSM beams in atmospheric turbulence. We start by introducing some basic parameters of GSM beams and numerical simulation approach [14, 15] in Section 2. Then, characteristics of beam spreading of PPCBs in free space and in turbulence are analyzed in Section 3, beam wandering and intensity scintillation are studied in Section 4. Finally, the influences of changing frequency of the random phase screen on the final intensity pattern are discussed in Section 5. And our results are summarized in Section 6.
2. Basic parameters of GSM beams and simulation approach
We assume that the optical field of a partially coherent beam at point ρ⃗ at the plane z = 0 is u(ρ⃗, 0; t), and the frequency-domain representation U(ρ⃗, 0; ω) can be given by
the cross-spectral density function is defined by 
where ρ = |ρ⃗| = (x 2 + y 2)1/2 and the asterisk denotes complex conjugation.
As to a GSM beam with partial spatial coherence, the cross-spectral density function can be written as
where I 0(ρ⃗) is the averaged intensity and is the spectral degree of source coherence. I 0(ρ⃗) and could be expressed as
where W 0 is the beam radius and lc (sometimes is called lateral correlation radius) determines the spectral degree of source coherence. For a spatially fully coherent beam lc would be infinite, and the degree of source coherence may degrade as lc decreases.
According to the random phase screen approach, the random phase screen that models the partial coherence is assumed to be ξ(x, y;t), then optical field of the source could be written as
where u 0(x, y,0) is optical field of the corresponding fully coherent beam. As to a GSM beam, the random phase screen can be modeled as 
where r(x,y;t) is a spatially uncorrelated random signal with standard deviation σr, f(x, y) is a Gaussian function with standard deviation σf,
and ⊗ indicates a spatial convolution. σr and σf satisfy the following equation
We use the approach described in [14, 15] to simulate the propagation of PPCBs. In free space, number of M independent random phase screens are generated by using equation (7). For the i-th realization, ξ(x, y;t). (i = 1,2,3……M) is used to create a source optical field according to equation (6). Then the propagation of the optical field is simulated by using FFT algorithm  and a resulting optical field at the detector plane is obtained. Finally, an M-average of the resulting fields creates one simulation optical field. The simulation field is more identical to the analytic result of a GSM beam as more resulting fields are averaged. Generally, 2000 realizations are enough to get a satisfied optical field.
In turbulent atmosphere, according to the concept of PPCB, we assume that during the period that atmospheric turbulence has changed once, random phase screen ξ has changed for K times. K is the relative changing frequency of ξ compared to turbulence. So firstly we generate K independent random phase screens ξ(x,y;t)i (i = 1,2,3……K). Then these K realizations of propagation in turbulence are similar to those in free space. We use the multiple phase-screen method [16–18] to simulate these propagation processes. Atmospheric turbulence along the path is also modeled by turbulent phase screens whose structure functions obey the well-known “2/3 law” of Kolmogorov spectrum. It should be pointed out that turbulent phase screens should not change in all these K realizations in order to keep atmospheric turbulence “frozen” during the time that ξ has changed for K times. Analogously, K resulting fields are averaged to obtain one simulation field. For another simulation, another K independent ξ should be created and the turbulent phase screens along the path are different from the previous ones. Then the simulation process is repeated to get hundreds or thousands of simulation fields which are averaged to analyze statistically the propagation effects of PPCBs.
3. Beam spreading of PPCB
The beam spreading of a partially coherent beam is different from that of a fully coherent beam. In this part, we study the characteristics of beam spreading of spatially pseudo-partially coherent GSM beams propagation in free space and in turbulent atmosphere respectively.
3.1 In free space
Figure 1 shows the comparison of the normalized intensity distribution of GSM beams with different lc and that of the fully coherent Gaussian beam, where the following parameters are adopted, such as wavelength: λ = 0.785 μm, beam radius: W 0 = 8 cm, propagation range: L = 1 km, and I is optical intensity and I max, CB is peak intensity of the fully coherent Gaussian beam. This figure shows that, as the degree of source coherence degrades, the distribution of intensity becomes more dispersive, and the peak intensity decreases simultaneously.
In Fig. 2 beam radius Wf scaled by W f,CB and peak intensity I max scaled by I max,CB are presented as a function of lc. It can be noticed that beam radius increases as lc decreases, and there is a significant reduction of peak intensity when lc becomes very short. When lc = 0.3 cm, peak intensity decreases to the 33.6 % of I max,CB. A comparison is also drawn between simulated results and analytical ones, and a perfect agreement can be observed clearly. From Fig. 1 and Fig. 2 we can conclude that free-space spreading of a partially coherent beam is larger than that of the fully coherent beam. This is an important property of partially coherent beams propagation in free space.
3.2 In turbulent atmosphere
A Kolmogorov power-law spectrum is assumed, and other parameters are chosen as: correlation radius of source lc = 0.6253 cm, wavelength λ = 0.785 μm, beam radius of source W 0 =8 cm, structure constant of refractive index Cn 2 =1×10-14 m-2/3, propagation range L = 3 km and relative changing frequency K=10. Figure 3 displays the behavior of beam radius in turbulent atmosphere and also in free space at different propagation distance. It can be clearly seen that beam radius of the PPCB is always larger than that of the CB no matter in turbulence or in free space. It might be affirmed that a PPCB always has a larger beam spreading than a CB.
However, from Fig. 3 one can also notice that beam radius of the PPCB in turbulence is nearly the same as that in free space, whereas beam radius of the CB in turbulence is obviously changed in comparison to that in free space. In order to describe clearly the beam spreading due to atmospheric turbulence, we divide beam radius in turbulence Wt by beam radius in free space Wf to get Wt/Wf which is called relative beam spreading. In some sense, relative beam spreading could reflect to what extent a beam is affected by atmospheric turbulence. Figure 4 shows the comparison of relative beam spreading between the PPCB and the CB. Relative beam spreading of the CB is much larger than that of the PPCB, especially at far propagation distance. This indicates that partially coherent beams are less affected than fully coherent beams by atmospheric turbulence.
4. Beam wandering and scintillation of PPCB
Here we also set K= 10, and choose lc=1.0cm, λ = 1μm, W 0=5cm, Cn 2 =1×10-13 m-2/3. Figure 5 shows the comparison of beam wandering RSM as a function of Rytov index β 0 2 between a PPCB and a CB. β 0 2 is defined as β 0 2 =1.23Cn 2 k 7/6 L 11/6, where k = 2π/λ is the wave number in vacuum. It can be observed that beam wandering RSM of the PPCB is approximately equal to that of the CB. Although beam wandering RSM of the CB is larger than that of the PPCB at large Rytov index, the difference is still very little. The degradation of degree of source coherence has no evident influence on beam wandering. And the behavior of beam wandering is almost independent of the source coherence.
Figure 6(a) and 6(b) display scintillation index as functions of Rytov index and propagation distance respectively. All parameters involved are the same as those in Fig. 5. From these two figures we can notice that scintillation index of the PPCB is obviously lower than that of the CB, and increases less rapidly with the increase of Rytov index and propagation distance. The scintillation saturation of the CB can be seen clearly when beam propagates to about 1.5km and peak value is nearly 2. However, scintillation index of the PPCB is lower than 1 along whole propagation path. It means that intensity fluctuations tend to decrease when beam source becomes less coherent. PPCBs are generally more resistant to the distortions caused by turbulent atmosphere than CBs.
5. Influence of relative changing frequency K
It was widely suggested that a partially coherent beam could be obtained by passing a fully coherent beam through a rotating random phase screen. The rotating speed reflects the changing frequency of the random phase screen. As it was referred in Section 2 that K is the relative changing frequency of random phase screen compared to atmospheric turbulence. The variation of K may have some effects on the optical field at the detector plane.
Figure 7 and Fig. 8 show beam radius and beam wandering RSM of PPCBs as a function of Rytov index for different K respectively. Here parameters of source partial coherence and turbulence are chosen to be the same as those in Fig. 6. It is easy to be detected that beam radius does not change with K, which means that beam spreading of a PPCB is independent of the changing frequency of random phase screen. It is also apparent in Fig. 8 that the change of K has some influence on beam wandering of the final intensity pattern. But the change of beam wandering is so little that we can say that beam wandering of a PPCB is not much sensitive to the changing frequency of random phase screen.
To analyze the changing frequency’s effects on beam intensity fluctuations, we choose K to be 1, 3, 5, 10 and 20 respectively. Figure 9 gives the scintillation index of the PPCB as functions of Rytov index and K, and we also illustrate the scintillation index of the corresponding CB. As we know that, each of the source optical fields created by different and independent ξ propagates differently in turbulent atmosphere. And source energy is redistributed at the detector plane for each realization. Thus a simulation intensity obtained from the average of K realizations appears smoother than a resulting intensity pattern without modulation of ξ. The intensity fade region caused by turbulence may be filled by the modulation, which results in a significant reduction of scintillation index when K changes from 1 to 3. However, with the increasing of K, decrease of scintillation index becomes slower. As it is shown that there is only a little difference of scintillation index when K changes from 10 to 20. Actually, once the changing frequency grows to a certain high level, the propagating properties of a PPCB are close to those of a PCB, which brings a saturation of scintillation index.
From Fig. 9 we can also see that scintillation index of the PPCB is even greater than that of the CB when the changing frequency of random phase screen equals to that of atmospheric turbulence (K=1). This is because that the random phase screen can also cause intensity fluctuations just like turbulent phase screens do. According to the process of generating a PPCB, we can deduce that the random phase screen appears “frozen” compared to atmospheric turbulence when K=1. On this occasion, the random phase screen acts like a single turbulent phase screen, which indeed causes additional intensity fluctuations. When K is greater than one, the combined effects of averaging process referred in Section 2 and the degradation of degree of source coherence make the intensity fluctuations decrease rapidly.
In this paper we have studied the propagation effects of spatially pseudo-partially coherent Gaussian Schell-model beams in turbulent atmosphere by using numerical simulation, and analyzed the characteristics of beam spreading, beam wandering and intensity scintillation. It was found that beam radius of a PPCB is greater than that of the corresponding CB both in free space and in turbulent atmosphere, which means that a PPCB has a larger spreading than a CB. However, the relative beam spreading of a PPCB is always smaller than that of the CB in turbulent atmosphere. And the degradation of degree of source coherence may cause a significant reduction of scintillation index. Thus, from the point of view of relative beam spreading and intensity fluctuations, PPCBs are less affected by atmospheric turbulence than CBs. This is a potential virtue of PPCBs that could be utilized to improve the performance of laser engineering. On the other hand, lager spreading of a PPCB may bring larger divergence of energy than the CB, which is not expected for the requirement of high power density. So, there is a balance between larger spreading and lower scintillation. The usefulness of PPCBs should depend on the particular application occasions. The results of beam wandering indicated that degradation of degree of source coherence does not have an evident effect on beam wander. There is no need to take beam wandering as a serious consideration for the selection of optimal light source between PPCBs and CBs.
Finally, we have talked about the influences of changing frequency of random phase screen on the final intensity pattern. Our results showed that the variation of changing frequency has no effect on beam spreading, and beam wandering is also not much susceptible to it. When changing frequency equals to that of atmospheric turbulence, the intensity fluctuations of PPCBs are even lager than that of CBs because of distortion induced by random phase screen. However, the increase of changing frequency may cause a significant reduction of intensity fluctuations. And there is no further reduction once the frequency grows to a certain high level.
The authors are very thankful to the reviewers for valuable comments. This work was supported by the National Natural Science Foundation of China (NSFC) under grant 0704071 and the Innovation Foundation of the Chinese Academy of Sciences under grant CXJJ-249.
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