1. Introduction

Partially coherent beams (PCB) are promising for free-space optical communication (FSOC) because of the reduced scintillations they can provide as compared to a fully coherent beam [1–5]. It was demonstrated that a PCB can be easily generated [6–9] in a multimode fiber (MMF) and modulated at gigabit-per-second (Gbps) data rates [10]. Multimode fiber bundles have also been explored [11, 12] for PCB generation, although their practical implementation is rather challenging. Compared to the existing multibeam FSOC systems [13, 14] a PCB-based system can be designed to be substantially simpler, cheaper and electronics-free in the optical antenna head [15]. The fact that a PCB diverges slightly faster than a fully coherent beam of the same aperture can be thought of as an advantage, reducing the need for pointing and tracking considerations in practical applications. All this justifies our continuing study in the properties of the MMF-based PCB approaches.

MMF PCB is generated simply by propagating light from a source with a certain spectral width, such as a superluminescent diode, through a length of a multimode fiber under the condition that intermodal dispersion exceeds the coherence time of the source [8]. In the ideal case when intermodal delay is infinitely larger than coherence time (inversely proportional to bandwidth) of the source the fiber modes no longer interfere at the output of the MMF. The output of the MMF is a PCB with its coherence radius $r_c=\lambda /\left(\pi NA\right)$ defined by the numerical aperture NA of the MMF. For practical source bandwidths and MMF lengths, however, the intermodal delay is, in fact, comparable to the source coherence time. In this case the coherence between any two points separated by a distance $\mathrm{\Delta }r\gg r_c$ may be larger on average than what it would have been for the case of non-interfering modes, Fig. 1. This residual coherence depends on the dimensionless ratio of the total modal dispersion to the coherence time of the source and can be controlled by adjusting input light source bandwidth or fiber length, or both [16, 17]. Interestingly, the degree of global coherence [18, 19] of such MMF PCB, as quantified, e.g., by the effective number of modes [20–22], is controlled by this residual coherence [23], and not by the coherence radius, as is the case for standard Gauss-Schell or “Jinc” models [24]. This decouples the angular divergence of MMF PCB from its global degree of coherence making experiments easier to perform and interpret.

 

Fig. 1 Interferometrically measured average modulus of the complex degree of coherence as a function of image shear Δr (spatial displacement between two images produced by the interferometer) of the output face of the fiber. Three color curves correspond to three different source bandwidths. Black dashed curve is the often used model 2|J₁ (k₀ NAΔr)/(k₀ NAΔr)|, where J₁ is the Bessel function of the first kind. Speckle pattern images of the fiber end face are shown on the right for each of the three curves in the main plot with speckle contrast values indicated.

Download Full Size | PDF

For FSOC applications the residual coherence needs to be minimized in order to maximize the benefits of the PCB for scintillation suppression. However, complete elimination of residual coherence would require impractical source bandwidth and fiber lengths. Thus, a question arises regarding the influence of the residual coherence on the overall scintillation index of the MMF PCB and what levels of residual coherence can be tolerated for practical FSOC applications. In this work we examine the effects of the residual coherence on the scintillation index in experiments in a laboratory turbulence and fit the data using the recently developed simple theoretical model [23] based on the view that MMF PCB can be considered as a combination of a coherent speckle and “ideal” homogeneous PCB described, e.g., by a “Jinc” degree of coherence, which scintillate almost independently. Using this model simple design criteria for a FSOC system are developed.

2. Description of the problem

The output from a stationary MMF excited by a light source with finite spectral width comprises a PCB characterized by a non-homogeneous non-Schell degree of coherence $\gamma \left({\overrightarrow{r}}_1,{\overrightarrow{r}}_2\right)$, where ${\overrightarrow{r}}_1$ and ${\overrightarrow{r}}_2$ are the two coordinates on the output endface of the fiber [8]. Averaging this function over many point pairs for a given spacing $\mathrm{\Delta }r=\left|{\overrightarrow{r}}_1-{\overrightarrow{r}}_2\right|$ results in a function, similar to those shown in Fig. 1, which were experimentally measured using interferometry [7, 8]. The near-constant wings in these functions comprise the average residual coherence, ${\gamma }_{\text{RC}}$, meaning that the electromagnetic fields at two points separated by more than a couple of r_c are partially coherent. This is in contrast to standard GSM or “Jinc” models (black curve in Fig. 1) for which the wings drop to zero rapidly.

MMF PCBs with residual coherence have speckle pattern intensity distributions (shown on the right of Fig. 1) characterized by speckle contrast $C={\left(〈I^2〉/〈I{〉}^2-1\right)}^{1/2}$ [25] directly corresponding to the residual coherence values [16, 17]: It was recently shown that the speckle contrast defined via averaging over random speckle patterns in the ensemble of fiber bends and twists (FBT ensemble) is very nearly equal to the residual coherence [17]. This result is significant in that it allows to deduct the value of the residual coherence via a much simpler measurement of FBT speckle contrast. Thus, in the following the terms “residual coherence” and “speckle contrast” will be used interchangeably.

Similar to the ideas from classical theory of coherence [26] the MMF PCB with finite residual coherence can be represented as a sum of a fully coherent part and an “ideal” PCB characterized by the “Jinc” degree of coherence and uniform intensity [23]. The coherent part is a fully developed speckle pattern with maximum speckle contrast, depending on polarization content, and average intensity I₁. The “ideal” PCB has intensity I₂ and so the intensity of the full beam is then the sum of the intensities of these two parts $I=I_1+I_2$. The intensity ratio $\xi =I_1/I_2$ can be related to the average residual coherence ${\gamma }_{\text{RC}}$ as $\xi ={\gamma }_{\text{RC}}/\left({\gamma }_{RC,max}-{\gamma }_{\text{RC}}\right)$, where ${\gamma }_{RC,max}$ is the maximum value the residual coherence can have, which is 1 for polarized beam and $\sqrt{0.5}=0.707$ for non-polarized case (in the experiments described below a nearly non-polarized case is realized). It can be seen that for one limit case when ${\gamma }_{\text{RC}}={\gamma }_{RC,max}$ we have $\xi \to \infty$, intensity of the PCB part $I_2=0$ and the beam is fully coherent. In the other limit ${\gamma }_{\text{RC}}=0$, ξ = 0 we have the opposite case $I_1=0$ and the beam is the “ideal” PCB with zero residual coherence.

Our goal is to derive the scintillation index of the full PCB with non-zero residual coherence upon propagation through turbulent atmosphere and relate it to the scintillation index values of the fully coherent beam and the “ideal” PCB, their intensity ratio as defined above, and their intensity fluctuation correlation coefficient. From the definition of the scintillation index $SI=〈I^2〉/〈I{〉}^2-1$ it is easy to show that

Similar ideas of separating coherent and incoherent parts in problems of polychromatic speckle scintillation in remote sensing geometries have been developed in the past [27]. In that work the coherent-incoherent separation emerges directly from the derivation of speckle propagation through turbulence using Huygens-Fresnel theory. Certain limiting approximations were naturally employed and experimental fitting was done to the dependence on turbulence strength, but not the speckle contrast. Studies of speckle contrast dependence were reported in [28] using extended Huygens-Fresnel principle and comparing to experimental results for a single value of contrast. Here we follow more of a phenomenological approach, not restricted by any specific turbulence propagation model and related approximations associated with it. We also fit the data to the dependence on speckle contrast for a given strength of a stationary laboratory turbulence.

For the turbulence which is not too weak we have $S{I}_1=0.5$ [28, 29] for non-polarized case. This is because to first order the effect of the turbulence on a speckled beam is to cause the motion of the speckles on the detector and the resulting scintillation index is just a square of the speckle contrast according to the definitions of these parameters. In the following section we experimentally determine the values SI, $S{I}_1$, $S{I}_2$, γ, and δ and verify the validity of this simple model.

3. Experiment

A stationary and well-controlled laboratory turbulence provides a suitable testbed to perform measurements of the scintillation index of a MMF PCB as a function of residual coherence, or equivalently the FBT speckle contrast. Our apparatus [9] comprises a large rectangular box across shorter dimension of which air is drawn by a set of fans, Fig. 2. Incoming air is both heated and chilled by traversing heater and cooler elements disposed at the air entry side of the box one atop the other. The optical beam traverses the box multiple times in a zig-zag pattern along the longer dimension of the box reflecting from large mirrors located at the opposite sides. Angular adjustment of mirror M₁ controls the number of passes the beam makes through the box. Turbulence inner scale on the order of 5 mm and index structure constants of up to $C_n^2={10}^{-10}$ m${ }^{-2/3}$ are typically obtained. At the output window of the box a 50-micron core MMF couples a portion of the light into an InGaAs detector (NewFocus, model 2053) the output of which is sent to a data acquisition system.

 

Fig. 2 Schematic of the experimental setup: SLD – superluminescent diode, SML – single-mode laser, EDFA – erbium-dopped fiber amplifier, MMF – PCB-generating multimode fiber, OSA – optical spectrum analyzer, MS – mode scrambler, FM – flipper mirror, DET – fiber-coupled InGaAs detector, CCD – InGaAs charge-coupled device, DAQ – data acquisition system. All fiber pigtails are single-mode.

Download Full Size | PDF

The front end of the experiment consists of two different light sources with adjustable bandwidths, Erbium-dopped fiber amplifier and a 105-micron core diameter MMF (Thorlabs AFS105, 2-meter length, $NA=0.22$), the output of which is collimated into free space with an aspheric lens before entering the turbulence chamber. The optical bandwidths of the two sources used are adjusted differently: The output of the broadband suplerluminescent diode (SLD, Thorlabs SLD1005) is coupled into a free-space spectral filter based on a grating-lens-slit-mirror setup, similar to a spectrometer or a pulse shaper. Spectral width range covering 0.5-20 nm (after EDFA) are obtained with this configuration by adjusting the slit opening. The second source is a semiconductor single-mode laser (JDSU CQF935), which lasing frequency weakly depends on the drive current. Modulating the laser current at a frequency higher than the maximum turbulence rate (200 Hz) allows effective broadening of the bandwidth in the amount dependent on the amplitude of the drive signal. By modulating the current with a sawtooth control signal we are able to achieve up to 1 nm effective bandwidth from this source. Thus, several orders of magnitude of overall bandwidth control is possible with the present setup. The corresponding values of the residual coherence range from the maximum value of about 0.7 down to 0.05, Fig. 3 (left).

 

Fig. 3 Left: Speckle contrast as a function of optical source bandwidth at the output of a 105-micron diameter core MMF, 2-meters in length. Right: Dependence of the scintillation index of a PCB in a laboratory turbulence on the FBT speckle contrast. The experimental data (symbols) is fitted by Eq. (1) (red curve).

Download Full Size | PDF

The two optical sources are combined on an Add/Drop filter, the output of which is coupled to the EDFA input. At the EDFA output a flipper mirror FM₁ can send the beam to an optical spectrum analyzer (OSA) or into the MMF for PCB generation. For optimal modal excitation in the MMF we use a mode scrambler near the input. The output from the MMF is collimated with a 25-mm focal length aspheric lens. A second flipper mirror FM₂ is used to either send the PCB into a turbulence chamber or for imaging with an InGaAs CCD (Xenics) for speckle contras measurement.

A fully spatially coherent beam was also used in the experiments for fine-tuning of the turbulence parameters via fitting the scintillation index to the Rostov theory [3, 4, 30]. A coherent beam is obtained simply by replacing the MMF with a single-mode fiber. We adjusted the turbulence to be in the weak/moderate range with Rytov parameter ${\sigma }_R=1.23C_n^2{k}^{7/6}{L}^{11/6}\le 0.5$ by appropriately setting the heater/cooler elements temperatures. The scintillation index of the quasi-Gaussian coherent beam was measured to be 0.2. The number of passes through the turbulence chamber was set to 9 so that the total propagation distance is near the transition length $L_{tr}=4\pi r_{c}a/\lambda$, where a is the radius of the beam entering the turbulence box [31].

In the main experiment we measure the scintillation index of the PCB as a function of the optical source bandwidth. Simultaneously, we measure the FBT speckle contrast at the output of the MMF by transferring its image onto the CCD with the help of a flip mirror FM₂, Fig. 2. The left panel of Fig. 3 shows the dependence of the speckle contrast on the source bandwidth. The data displays the expected behavior [16, 17, 32], with FBT speckle contrast decreasing roughly as a square root of optical source bandwidth for large bandwidths.

The right panel of Fig. 3 plots the scintillation index of the PCB as a function of speckle contrast and represents the main finding of the present work. As expected, we observe that for large values of contrast the scintillation index is close to the square root of contrast, meaning that the main effect of the turbulence is to agitate the speckles on the detector. The PCB is now understood to consist mainly of a coherent beam with $\xi \to \infty$ and $SI=S{I}_1$, see Eq. (1). As we increase the source bandwidth (reduce FBT speckle contrast) the scintillation index decreases past the value for the coherent beam with a near-flat wavefront (slightly diverging Gaussian beam) (0.2) and settles on a value $\approx 0.08$—the scintillation index of the “ideal” PCB $S{I}_2$. One practical point here is that speckle contrast reduction below the reasonable value of 0.1-0.2 is not required to substantially improve performance. This is an important observation because speckle contrast is a rather slow function of the product of bandwidth and fiber length, or equivalently the ratio of the modal dispersion to the coherence time of the source $\tau /t_c$. Since this parameter defines the pulse stretching due to modal dispersion its value is limited from above by the required data rate in the communication channel [10].

In addition to the $S{I}_1$ and $S{I}_2$ values the fit in Fig. 3(right panel) requires one more parameter—correlation between the coherent and the “ideal” PCB parts of the full PCB. This parameter δ was measured in a similar setup with a slightly modified data acquisition method. Specifically, we current-modulated the SLD and externally modulated the SML at two different frequencies of 100 kHz and 240 kHz in order to separate them on the detection side. The detected signal is digitized at 1 MS/s and a fast Fourier transform is computed every millisecond so that the Nyquist frequency of 500 Hz is still larger than the maximum turbulence rate on the order of 200 Hz. The time-series data at 1 kHz from the two laser sources are then correlated in software. The result of this measurement is that the correlation coefficient δ is rather small, not exceeding 0.1, indicating that the coherent and the“ideal” PCB parts of the complete PCB scintillate almost independently. All the measured parameters are used in the fit of Fig. 3 (right panel), and the reasonable agreement observed suggests the validity of the model assumed.

4. Discussion

For a given FSOC channel the scintillation index $S{I}_2$ of an “ideal,” PCB can be calculated using a number of existing models [1, 2, 4, 31]. The result will be a complicated function of multiple parameters, such as propagation distance, beam aperture radius, beam divergence, wavelength, as well as turbulence parameters, such as index structure coefficient, inner and outer scales, etc. Nevertheless if $S{I}_2$ can be obtained the present model allows to easily estimate the intensity ratio ξ, average residual coherence ${\gamma }_{\text{RC}}/{\gamma }_{\text{RC},max}=\xi /\left(1+\xi \right)$ and from there the required $\tau /t_c$ in order to generate MMF PCB that would produce the full-beam scintillation index not too much exceeding $S{I}_2$. The way this is accomplished is by setting the left-hand side of Eq. (1) to $\alpha S{I}_2$, where α is a number slightly exceeding 1 representing the tolerable excess of SI over the required $S{I}_2$, say 10% ($\alpha =1.1$). Then Eq. (1) becomes a quadratic equation for ξ, the positive root of which gives the value of ξ required to achieve $SI=1.1S{I}_2$.

 

Fig. 4 Scintillation index of the full beam as a function of coherent-to-PCB power ratio ξ (a) and average residual coherence γ_RC (b) for various values of the scintillation index of the “ideal” PCB SI₂ in turbulence (solid black curves). From top to bottom curve the SI₂ values are 0.1, 0.05, 0.03, 0.01, 0.005, and 0.003. Colored symbols represent SI = 1.1 SI₂ values with vertical lines shown to help read the solution values from the horizontal axis in each plot.

Download Full Size | PDF

An illustration of this simple procedure is given in Fig. 4. A symbol on each curve represents scintillation index of a full beam 10% higher than that of the “ideal” PCB. The curves are computed using Eq. (1) and several different $S{I}_2$ values ranging from 0.1 to 0.003. The vertical color lines dropped to the horizontal axis in each plot help read off the values of ξ or ${\gamma }_{\text{RC}}$ required to obtain these values. In our experiments described above $S{I}_2$ was near 0.08 and from Fig. 4(b) we deduce that a reasonably close values of full beam SI can be achieved with ${\gamma }_{\text{RC}}$ between 0.1 and 0.2, as was experimentally established in the previous section. From Fig. 3(a) it follows that these values can be obtained in a 2-meter long MMF with source bandwidth of a few nanometers. Other experimental conditions can be treated in a similar manner.

5. Conclusion

Scintillation index of a partially coherent beam generated in a multimode fiber is measured in a laboratory turbulence as a function of the residual coherence, or equivalently speckle contrast evaluated in the ensemble of random bends and twists of the fiber, controlled by the input source bandwidth. When the speckle contrast is large the turbulence agitates the speckles on the detector and the resulting scintillation index is equal to the square of the speckle contrast. In the opposite case of low speckle contrast corresponding to the optical source with large bandwidth the scintillation index saturates at the value corresponding to the “ideal” PCB. A simple model for the full beam scintillation index is developed based on the notion that it can be represented by a coherent and incoherent parts with their relative intensities determined by the value of the residual coherence. These two parts are found to scintillate almost independently. A recipe is provided for determining the minimum fiber length and/or source bandwidth in order to obtain full beam scintillation index relatively close to the “ideal” PCB. Under the experimental conditions considered the required scintillation index is obtained with source bandwidth of a few nanometers using a 2-meter long 105 μm core diameter fiber.