1.

Introduction

The parameter that characterizes the mean radius of a Gaussian beam after propagation through optical turbulence over a distance $L$ from the source or transmitter is referred to as the long-term beam spread.^1–3 This parameter holds significance in various applications, particularly in directed energy scenarios. In the realm of free-space optical communications, it becomes essential to extend the theoretical modeling of the long-term beam spread to the focal plane, where an optical detector or an optical fiber is commonly positioned.⁴

This paper extends the analysis of the long-term beam spread and Strehl ratio (SR) in the focal plane, as explained in Ref. 4, to conduct an examination of other optical power-related important metrics for budget link calculations. In particular, we calculate the average power in the bucket (PIB) captured by the receiver aperture and, depending on the presence of a fiber or a sensor in the focal plane, the average power into the fiber (PIF) or the average power over the sensor (POS).

We perform this analysis for Gaussian beams with general geometries, encompassing focused, collimated, and divergent configurations. Our methodology relies on the ABCD matrix representation¹ and accommodates scenarios where turbulence can deviate from the assumptions of the Kolmogorov model.²

2.

Kolmogorov and Non-Kolmogorov Turbulence

Let us focus on the following power law-dependent non-Kolmogorov power spectrum:²

The schematic of a free space optical communication system is shown in Fig. 1. A Gaussian beam with spot radius $W_0$ exits the transmitter aperture with geometry defined by a couple of beam parameters $({\mathrm{\Theta }}_0=1-\frac{L}{F_0};{\mathrm{\Lambda }}_0=\frac{2L}{k{W}_0^2})$.

Fig. 1

Free space optical communication scheme.

Here $L$ is the path length, $F_0$ is the radius of curvature of the wavefront at the transmitter, and $k$ is the wavenumber. After propagation, the beam is captured at the receiver aperture by a collecting lens of diameter $D_G$. The beam on the collecting lens has geometry described by ${\mathrm{\Theta }}_1=\frac{{\mathrm{\Theta }}_0}{{\mathrm{\Lambda }}_0^2+{\mathrm{\Theta }}_0^2};{\mathrm{\Lambda }}_1=\frac{{\mathrm{\Lambda }}_0}{{\mathrm{\Lambda }}_0^2+{\mathrm{\Theta }}_0^2}$ and it is focused at distance $L_f$ behind the lens, where a detector or an optical fiber is located.

Note that here we consider a hard aperture with diameter $D_G$ related to the soft aperture Gaussian lens of radius $W_G$ by the known expression $D_G^2=8·W_G^2$ (see Ref. 1). The beam in the focal plane has the geometry $({\mathrm{\Lambda }}_2=\frac{L}{L_f}·\frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Omega }}_G};{\mathrm{\Theta }}_2=0)$, where ${\mathrm{\Omega }}_G=\frac{2L}{k{W}_G^2}$ and $F_G=L_f$ is the focal length of the collecting lens. The diffraction-limited beam spot radius, $W_2$ in the focal plane is related to the diffraction parameter ${\mathrm{\Lambda }}_2=\frac{2L_f}{k{W}_2^2}$, therefore, $W_2=\sqrt{\frac{2L_f}{k{\mathrm{\Lambda }}_2}}$.

Note that all the geometric parameters, ${\mathrm{\Theta }}_0,{\mathrm{\Lambda }}_0$; ${\mathrm{\Theta }}_1,{\mathrm{\Lambda }}_1$ and ${\mathrm{\Theta }}_2,{\mathrm{\Lambda }}_2$ are free space parameters, however they enter the equation describing the change in the on-axis mean irradiance in the focal plane after propagation in turbulence, Eq. (8).

3.

Long Term Beam Spread Analysis in the Focal Plane

The long-term beam spread is the parameter that physically describes the mean radius of the Gaussian beam after its propagation in optical turbulence. The long-term spot radius in the focal plane can be defined as in Refs. 1, 2, and 4

As previously introduced, to calculate $T_{\text{focal}}$ we use the ABCD ray-matrix representation shown in Ref. 1, and we neglect the effect of turbulence on the beam behind the lens (beam path from the lens to the detector). Under this condition, we use two statistical terms¹

For a beam focused on the photodetector/optical fiber we impose¹

Note that if $L_f=F_G$, $\frac{L}{L_f}-\frac{L}{F_G}+1-{\mathrm{\Theta }}_1=0$ gives ${\mathrm{\Theta }}_1=1$, which means collimated beam incidents the collecting lens. For the general case of a beam not arriving collimated on the lens $({\mathrm{\Theta }}_1\ne 1)$, to insure the focusing on sensor/optical fiber, the distance behind the lens has to be $L_f=\frac{F_G·L}{L+({\mathrm{\Theta }}_1-1)·F_G}$. We remark that Eq. (3) diverges using the spectrum of Eq. (1) because outer scale is supposed to be infinite. However, it can be shown that such a singularity is removed in Eq. (8) and a finite result is found.

3.2.

Non-Kolmogorov Turbulence Case

Using the non-Kolmogorov turbulence power spectrum, Eq. (1) we obtain

Eq. (13)

$$T_{\text{focal}}(\alpha )=\frac{1}{4}\frac{\alpha }{\sin (\alpha ·\frac{\pi }{4})}·{\tilde{\sigma }}_R^2(\alpha )·{({\mathrm{\Lambda }}_1+{\mathrm{\Omega }}_G)}^{1-\frac{\alpha }{2}}·{\int }_0^1{[(1-{\overline{\mathrm{\Theta }}}_1\xi {)}^2+{\mathrm{\Lambda }}_1{\mathrm{\Omega }}_G{\xi }^2]}^{\frac{\alpha }{2}-1}\mathrm{d}\xi ,$$

where ${\tilde{\sigma }}_R^2(\alpha )=-8{\pi }^2·\mathrm{\Gamma }(1-\frac{\alpha }{2})·\frac{1}{\alpha }·\sin (\alpha ·\frac{\pi }{4})·A(\alpha )·{\tilde{C}}_n^2·k^{3-\frac{\alpha }{2}}{L}^{\frac{\alpha }{2}}$ is the non-Kolmogorov Rytov variance defined in Ref. 2.

For a beam arriving collimated on the collecting lens, ${\mathrm{\Theta }}_1=1$

Eq. (14)

$$T_{\text{focal}}(\alpha )=\frac{1}{4}\frac{\alpha }{\sin (\alpha ·\frac{\pi }{4})}·{\tilde{\sigma }}_R^2(\alpha )·{({\mathrm{\Lambda }}_1+{\mathrm{\Omega }}_G)}^{1-\frac{\alpha }{2}}·{}_{2}F_1(1-\frac{\alpha }{2},\frac{1}{2};\frac{3}{2},-{\mathrm{\Lambda }}_1{\mathrm{\Omega }}_G).$$

For a plane wave incident on the collecting lens, $({\mathrm{\Lambda }}_1=0,{\mathrm{\Theta }}_1=1)$ and we obtain

Eq. (15)

$$T_{\text{focal},\mathrm{pl}}(\alpha )=\frac{1}{4}\frac{\alpha }{\sin (\alpha ·\frac{\pi }{4})}·{\tilde{\sigma }}_R^2(\alpha )·{\mathrm{\Omega }}_G^{1-\frac{\alpha }{2}}.$$

For a spherical wave incident on the collecting lens, $({\mathrm{\Lambda }}_1=0,{\mathrm{\Theta }}_1=0)$ and we obtain

Eq. (16)

$$T_{\text{focal},\mathrm{sph}}(\alpha )=\frac{1}{4}·\frac{1}{\alpha -1}\frac{\alpha }{\sin (\alpha ·\frac{\pi }{4})}·{\tilde{\sigma }}_R^2(\alpha )·{\mathrm{\Omega }}_G^{1-\frac{\alpha }{2}}.$$

Although we found expressions also for the non-Kolmogorov case, we show only results for Kolmogorov turbulence, $\alpha =11/3$.

For a specific set of parameters, we plot in Fig. 2 the term $T_{\text{focal}}$ as a function of the diffraction parameter, ${\mathrm{\Lambda }}_0=\frac{2L}{k{W}_0^2}$. Note that in our plots we change the spot size radius at the transmitter, $W_0$ from 0.001 m to 0.5 m covering the whole range from the near field ${\mathrm{\Lambda }}_0\ll 1$ to the far field ${\mathrm{\Lambda }}_0\gg 1$. The value ${\mathrm{\Lambda }}_0=1$ corresponds to the value of $W_0=\sqrt{\frac{2L}{k}}=\sqrt{\frac{\lambda L}{\pi }}$, which is the radius of the first Fresnel zone. Note also that for the case of a collimated beam, the value ${\mathrm{\Lambda }}_0=1$ corresponds to the Rayleigh distance, $z_R=\frac{1}{2}k{W}_0^2$, which forms a separation line between the near and far field. All the following plots will show only results for collimated and focused beams because of their relevance to free space communications. In terms of the formalism used in this paper, it should be clear that the collimated beam at the transmitter has a wavefront radius of curvature $F_0=\infty$; the focused beam has $F_0=L$. Note furthermore that $F_0=\infty$; ${\mathrm{\Lambda }}_0=0$ is equivalent to the plane wave case at the transmitter and $F_0=L;{\mathrm{\Lambda }}_0\to \infty$ is equivalent to the spherical wave case.

Fig. 2

Spread term $T_{\text{focal}}$ as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for a beam collimated at the transmitter (blue) or focused on the receiver (red) (Kolmogorov turbulence case). Distance is fixed to the value $L=2\mathrm{km}$.

We deduce from Fig. 2 that for a collimated beam and for this set of parameters, the spread term $T_{\text{focal}}$ is maximum in the near field. Note that in the region ${\mathrm{\Lambda }}_0\ll 1$ a collimated beam (${\mathrm{\Theta }}_0=1$) is essentially a plane wave. This explains why the spread is higher: the radial dimension of the beam geometrically intercepts a higher number of turbulence cells along the path. The minimum value of $T_{\text{focal}}$ is reached at about 2.5 times the Fresnel distance (${\mathrm{\Lambda }}_0=1$) and, for ${\mathrm{\Lambda }}_0\gg 1$, $T_{\text{focal}}$ increases up to the saturation value double than unity (it approaches the spherical wave case, ${\mathrm{\Lambda }}_0\to \infty ,{\mathrm{\Theta }}_1\to 0$). On the other hand, a focused beam shows a lower spread than a collimated one all over the range from the near field to the far field, where it approaches again the spherical wave case ${\mathrm{\Lambda }}_0\to \infty ,{\mathrm{\Theta }}_0=0$. As for the collimated beam case, this is physically explained by the radial dimension of a focused beam, which intercepts a lower number of turbulence cells along the path.

In Fig. 3, we plot, for the collimated beam case, two spot radii in the focal plane, respectively, the long-term beam spread $W_{\mathrm{LT},f}$ and the diffraction limited spot radius $W_2$, both of them as a function of the diffraction parameter, ${\mathrm{\Lambda }}_0=\frac{2L}{k{W}_0^2}$. We deduce from Fig. 3 that in the near field ${\mathrm{\Lambda }}_0\ll 1$ (where plane wave model holds well) the long-term beam spread in the focal plane is higher than in the far field ${\mathrm{\Lambda }}_0\gg 1$, where the spherical wave model holds. This explains the saturation of both plots as well. Note also that the maximum diffraction-limited (no turbulence) spot radius $W_2$ can be obtained when the receiver is located at Fresnel distance, ${\mathrm{\Lambda }}_0=1$. However, in turbulence it is more beneficial to locate the receiver in the far field to reduce the long-term spot size of the beam.

Fig. 3

Long-term beam spread term $W_{\mathrm{LT},f}$ (blue) and the diffraction limited spot radius $W_2$ (red) as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for a beam collimated at the transmitter (Kolmogorov turbulence case only). Distance is fixed to the value $L=2\mathrm{km}$.

This conclusion also applies to a focused beam as shown in Fig. 4. Here we plot the same parameters as in Fig. 3 but now for a beam focused on the receiver. Note how the value of the long-term beam spread in the focal plane is essentially driven by the diffraction parameter ${\mathrm{\Lambda }}_2=\frac{L}{L_f}·\frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Omega }}_G}=\frac{2L_f}{k{W}_2^2}$ or, in other words, it is anchored on the diffraction-limited spot size $W_2$.

Fig. 4

Long-term beam spread term $W_{\mathrm{LT},f}$ (blue) and the diffraction limited spot size $W_2$ (red) as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for a beam focused on the receiver (Kolmogorov turbulence case). Distance is fixed to the value $L=2\mathrm{km}$.

To highlight the differences between the two beam geometries, we plot again in Fig. 5 the long-term beam spread in the focal plane $W_{\mathrm{LT},f}$ as a function of the diffraction parameter ${\mathrm{\Lambda }}_0=\frac{2L}{k{W}_0^2}$ for both a collimated and a focused beam (these two plots are the blue curves shown also in Figs. 3 and 4). We deduce from Fig. 5 that in the near field a collimated beam shows a lower long-term beam spread than a focused beam. However, they approach the same value when ${\mathrm{\Lambda }}_0=1$ (Fresnel distance) or higher. In general, Fig. 5 suggests that independently from using a collimated or focused beam, to obtain the smallest long-term spot size in the focal plane the receiver should be located in the far field, ${\mathrm{\Lambda }}_0\gg 1$ (spherical wave model). We remark that in the far field Eq. (12) for spherical wave model holds well.

Fig. 5

Long-term beam spread term $W_{\mathrm{LT},f}$ as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for both beam geometries: collimated and focused (Kolmogorov turbulence case). Distance is fixed to the value $L=2\mathrm{km}$.

4.

Strehl Ratio Analysis

The SR as one of the most used metrics at the focal plane is defined as¹

For the spherical wave case $({\mathrm{\Lambda }}_1=0,{\mathrm{\Theta }}_1=0)$ and Kolmogorov turbulence, $\alpha =11/3$, Eq. (18) reduces to the same result shown in Ref. 1 (see page 623)

We plot in Fig. 6 the SR as a function of the diffraction parameter ${\mathrm{\Lambda }}_0=\frac{2L}{k{W}_0^2}$ for the same scenario of propagation as previous plots. We deduce from Fig. 6 that using a focused beam and a receiver aperture located in the near field, ${\mathrm{\Lambda }}_0\ll 1$, is beneficial in terms of SR with respect to use a collimated beam. Physical conclusions are the same as those of Fig. 2. (The beam geometry defines the cross section of the beam while propagating through turbulence and focused beam intercepts a smaller number of turbulence cells than a collimated beam.)

Fig. 6

SR as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for a beam collimated at the transmitter (blue) or focused on the receiver (red) (Kolmogorov turbulence case only). Distance is fixed to the value $L=2\mathrm{km}$.

5.

Power in the Bucket

The transmitted mean power of a Gaussian beam with peak intensity $A_0^2$ and spot radius $W_0^2$ is $P_{Tx}=\frac{1}{2}\pi A_0^2{W}_0^2$ [Watt]. The fraction of the mean power entering the receiver aperture after propagation (distance L) is the PIB.³ Considering a Gaussian lens (soft aperture) with radius $W_G$ related to the hard aperture diameter, $D_G$ by $W_G=\frac{D_G}{2\sqrt{2}}$ and supposing the turbulence affected beam still Gaussian,¹ the average $⟨\mathrm{PIB}⟩$ can be expressed as

For a large beam (close to a plane wave) incident on the lens, an approximation is³

We plot in Figs. 7 and 8 the percentage of PIB scaled by the transmitted power, $P_{Tx}$ as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a collimated beam at the transmitter or focused on the receiver (Kolmogorov turbulence case only). We deduce from Figs. 7 and 8 that the maximum PIB is reachable when the beam is focused on the receiver (see Fig. 8) and the collecting lens is located in the near field. Also, when the receiver is positioned in the far field the spherical wave model [please see Eqs. (12) and (19)] holds well and there is essentially no difference in using either beam geometry.

Fig. 7

Percentage of PIB scaled by the transmitted power as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a collimated beam at the transmitter (Kolmogorov turbulence case only).

Fig. 8

Percentage of PIB scaled by the transmitted power as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a beam focused on the receiver (Kolmogorov turbulence case only).

6.

Power over the Sensor

Similar to the PIB but now on the focal (detector) plane, the fraction of transmitted mean power illuminating the sensor/detector (POS) can be expressed as

Supposing the beam illuminating the sensor is consistently larger than the sensor itself (the Gaussian beam across the sensor is almost a plane wave), an approximated expression of Eq. (22) is

We plot in Figs. 9 and 10, respectively, the percentage of POS scaled by the transmitted power, $P_{Tx}$ as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a collimated beam at the transmitter and for a beam focused on the receiver (Kolmogorov turbulence case only).

Fig. 9

Percentage of POS scaled by the transmitted power as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a collimated beam at the transmitter (Kolmogorov turbulence case only).

Fig. 10

Percentage of POS scaled by the transmitted power as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a beam focused on the receiver (Kolmogorov turbulence case only).

We deduce from Figs. 9 and 10 that, for both geometries and for this specific set of parameters, the maximum POS is reachable when ${\mathrm{\Lambda }}_0$ assumes values in the interval from two to five times the Fresnel distance $({\mathrm{\Lambda }}_0=1)$, depending on path distances (note that the peak shifts slightly to the left at shorter distances). Also, there is essentially no difference of using the two different beam geometries (collimated or focused beam) in the far field (spherical wave model holds well in such a case).

Fig. 11

Percentage of PIF scaled by the transmitted power as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a collimated beam at the transmitter (Kolmogorov turbulence case only).

7.

Power into the Fiber

Supposing an optical fiber positioned in the focal plane, the mean power coupled into the fiber can be expressed as³

We plot in Figs. 11 and 12 the PIF as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a collimated beam at the transmitter or for a beam focused on the receiver (Kolmogorov turbulence case only). Similar to the POS in Fig. 9, we deduce from Fig. 11 (collimated beam) that, for this specific set of parameters, the maximum PIF is reachable when ${\mathrm{\Lambda }}_0$ assumes values in the interval from two to five times the Fresnel distance $({\mathrm{\Lambda }}_0=1)$, depending on path distances (note that the peak shifts slightly to the left at shorter distances).

Fig. 12

Percentage of PIF scaled by the transmitted power as a function of the diffraction parameter ${\mathrm{\Lambda }}_0$ for several distances of the receiver for a beam focused on the receiver (Kolmogorov turbulence case only).

Also, when the beam is focused on the receiver, we deduce form Fig. 12 that the maximum PIF (for this specific set of parameters) is reached mostly in the near field. Finally, similarly to previous metrics, we note that when the receiver is located in the far field the spherical wave model holds well and there is essentially no difference in using either beam geometry (collimated or focused beam case).

8.

Conclusion

In this paper, we used the ABCD matrix formulation to obtain the analytical expressions of the main optical power-related metrics for a Gaussian beam after propagation through atmospheric turbulence. Specifically, these optical power-related metrics are: the long-term beam spread in the focal plane, SR, PIB, POS, and PIF. We investigated those metrics as a function of the diffraction parameter, ${\mathrm{\Lambda }}_0=\frac{2L}{k{W}_0^2}$.

For a specific set of parameters, we found that both the maximum SR and PIB are reachable when a beam focused on the receiver is used and the collecting lens is located in the near field. In addition, we found that for a collimated beam the maximum POS and PIF are reachable when the diffraction parameter ${\mathrm{\Lambda }}_0$ assumes values in the interval two to five times the Fresnel distance $({\mathrm{\Lambda }}_0=1)$ depending on path distances (the POS and PIF peaks shifts slightly to the left at shorter distances). However, when the beam is focused on the receiver, the maximum PIF is mostly reached in the near field.

Finally, in the far field and for all metrics analyzed in this paper, the spherical wave model holds well and there is essentially no difference of using the two different beam geometries (collimated or focused beam). Our results can be useful for the budget link analysis for free space optical communications.