1. Introduction

Coherence properties of light are important in a wide range of applications, such as imaging [1], optical coherence tomography [2, 3] and free-space optical communication [4–6]. Spatially incoherent or partially coherent sources abound, but do not always possess the properties required for a particular application, such as high modulation frequency. Often times it may be desired to reduce the spatial coherence of a source to a predetermined value or as much as possible. Such decoherization is easily achieved for optical sources of finite spectral bandwidth by propagating in a multimode fiber (MMF) of an appropriate length [7]. Modal dispersion, fiber length, and the bandwidth of the source determine the spatial coherence properties at the MMF output [8].

Historically, multimode waveguides have been used to obtain uniform spatial illumination on a target [9] by reducing the speckle contrast. Reduced speckle contrast is indeed a good indication of spatial coherence, however more information can be obtained if the modulus of complex degree of coherence |γ| is measured. Additionally, a simple and universal model describing the characteristics of |γ| as a function of fiber and optical source parameters would be of great value.

Pask and Snyder [10] directly carried the Van-Citter Zernike theorem from free-space to a fiber by restricting the numerical aperture NA and expectedly obtained the familiar γ (Δr) = 2J₁ (k₀NAΔr)/(k₀NAΔr) result for totally incoherent (non-interfering) modes. Here J₁ is a Bessel function of the first kind, Δr is the distance between the two points on the fiber output surface and k₀ is the vacuum wavenumber. Although not confirmed experimentally this result is still used faithfully today [3]. Crosignani, et al. [11, 12] were the first to consider the influence of the source bandwidth on the spatial coherence properties at the output of a MMF using explicit expressions for the fiber modes in a weakly guided approximation. Their analytical results provide only little insight into the case of interfering modes, most focus being placed on the dependence on fiber length. A serious consideration of interfering modes excited by a broadband source was undertaken by Deryugin, et al. [13], where both spatial and temporal degrees of coherence were considered. The details of their calculations were not spelled out, however it was obvious that in a realistic case of interfering modes the simple Bessel form of Pask and Snyder [10] is no longer valid. Their theory was not carried further to arrive at any universal dependencies.

The fact that the output of a MMF is strongly inhomogeneous complicates the measurement of |γ| considerably. Spatial interferometry is a typical technique used to measure coherence properties of light [1]. Multiple experiments were conducted in the past to measure the spatial degree of coherence at the output of various fibers and waveguides. Dzhibladze, et al. [14] seem to be one of the first to attempt to measure |γ| at the output of a MMF. They used the classic double-slit Young’s setup to obtain spatial fringes from a short large-core MMF excited by a HeNe laser. The slit spacing was fixed so only one data point was obtained for each fiber sample. Nevertheless, convincing length-dependence of the residual coherence—the value of |γ| outside the main peak of the spatial correlation function—was obtained. Spano [15] employed an image-flipping interferometer [16] to measure |γ| at the output of a couple of guided structures which supported a small number of modes with the goal of determining the modal content of the light. It was shown that |γ| is a much more sensitive metric of the relative mode amplitudes than the intensity I (r). The influence of the source bandwidth on |γ| at the output of a few-mode MMF was considered theoretically and experimentally by Imai and Ohtsuka [17,18]. In their work modal interference was taken into account with explicit expressions for a few low-order modes used in numerical computation of the complex degree of coherence.

Spatial coherence was measured recently for other important light sources and using other interferometric techniques. Sheering Sagnac interferometer [19] was used to measure spatial coherence properties of vortex beams [20] and sunlight [21]. Sheering Sagnac interferometer, however, does not allow the measurement of γ, but rather the correlation function Γ, the two being related to each other through normalization to individual intensities of the two interfering optical fields [1]: ${\gamma }_{12}(\tau )={\mathrm{\Gamma }}_{12}(\tau )/\sqrt{I_1{I}_2}$, where τ is the delay between the two fields [22]. It can be seen that for an inhomogeneous source, such as MMF, for which I₁ and I₂ are functions of spatial coordinates (points 1 and 2), γ and Γ will have vastly different spatial dependences. It is, in fact, the modulus of the complex degree of coherence |γ| which is the proper measure of coherence, and not the correlation function [23].

For a complicated inhomogeneous source, such the output of a MMF pumped by a broadband optical source, to properly describe the spatial coherence properties of light one needs to specify the complete function |γ₁₂|, or at least the coherence radius—half width of the central peak— and the value of the average residual coherence. In this work we use a lateral-sheering, delay-dithering Mach-Zehnder interferometer [24] to measure |γ| at the output of a step-index MMF pumped with optical sources of various spectral widths. For monochromatic pumping the output of the MMF has a high-contrast speckle structure, but |γ| = 1 for all pairs of points. On the other hand, for broadband pumping the output is partially coherent. The coherence radius is determined by the NA of the fiber, while the residual coherence will be shown to depend on the source bandwidth and the number of non-degenerate modes in the fiber. We also develop a simple intuitive model to numerically compute the averaged residual coherence 〈|γ|〉 as a universal function of two dimensionless parameters—the ratio of the total modal dispersion to source coherence time and the number of modes.

2. Experiment

The interferometer used in this work is described in detail elsewhere [24]. Briefly, the interferometer consists of two arms with corner-cube retroreflectors in each arm, Fig. 1. The output face of the MMF studied is positioned at the “Obj.” plane and its intermediate image is produced with magnification before the interferometer by an afocal system consisting of lenses L₁ and L₂ having different powers. The intermediate image is then propagated through the interferometer and two final images are then formed by the lens L₃—one from the vertical arm and the other from the horizontal arm. The image from the horizontal arm is stationary, while the one from the vertical arm can be scanned (sheered) across the first by laterally translating the corner cube in that arm. A small detector therefore receives light from two separate points P₁ and P₂ on the MMF’s output face. The retroreflector in the horizontal arm is controlled axially with another translation stage for coarse delay adjustment, and is also mounted in a piezo-controlled kinematic mount for rapid delay dither over a range of a few interference fringes. The interferometer performs three measurements required to compute |γ| nearly simultaneously: temporal interferogram produced by rapid delay dither of the horizontal-arm retroreflector, and two signal intensities I₁ and I₂, from points P₁ and P₂ from which the modulus of the complex degree of coherence can be computed [24].

 

Fig. 1 Schematic of the lateral-sheering, time-dithering MZ interferometer. Principal rays from object points P₁ (red) and P₂ (green) are shown to split at the first surface of the beamsplitter into horizontal (solid) and vertical (dashed) arms thus creating two displaced images on the Detector Plane for each point. The image of P₂ through the vertical arm and the image of P₁ through the horizontal arm overlap and interfere as shown on the right.

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The fiber used in our experiments is a step-index MMF with 105 μm core diameter (Thorlabs AFS105) with numerical aperture NA = 0.22 and estimated modal dispersion τ_mod ≈ L NA²/2nc of about 50 fs per millimeter of length. We used two different fiber lengths L of 15 cm and 5 m in our experiments to cover a wide range of the dimensionless parameter Δt/t_c, where Δt = Lτ_mod is the total modal delay and t_c is the coherence time of the input source as described below.

To pump the MMFs we used several arrangements. For the monochromatic source a CW semiconductor laser diode with 1 MHz linewidth (JDSU CQF935) was used. For broadband pumping with variable optical bandwidth we used a system comprising a broadband optical source (Amonics ALS-CL-23-B-FC), a pulse shaper with a variable slit in its Fourier plane, and an Erbium-doped fiber amplifier (Optilab EDFA-1-23-B). The pulse shaper is capable of narrowing the input spectrum to about 0.5 nm. Subsequent amplification in EDFA brings the signal power back to 200 mW with amplified spontaneous emission (ASE) level of 25 dB below the signal at the worst. The widest spectrum of about 40 nm at 10% intensity can be obtained by blocking the input into the EDFA and using its ASE directly. For each experiment the source spectral width is measured with an optical spectrum analyzer and converted to coherence time t_c. To excite a large number of modes we overfill the acceptance angle of the MMF by tightly focusing the initially fully spatially coherent light on the input face of the MMF [25]. The position of the focus is not centered on the core of the MMF in order to excite all possible skew rays in the fiber. Furthermore, in some experiments a mode mixer was used near the input, however no difference was observed in the results, indicating that both methods yield rich modal content.

Pumping the MMF with a monochromatic source results in a highly non-uniform (speckle) intensity distribution, while the measured modulus of the complex degree of coherence |γ| as a function of lateral sheer Δr is equal to unity throughout the whole sheer range with about a percent random error, Fig. 2 (black crosses). For broadband pumping the output will have nontrivial coherence properties, which depend on the dimensionless ratio Δt/t_c and the number of modes. The output will be substantially incoherent when the number of modes is large and each pair of modes disperse by more than t_c, however little is known about the transition region between the coherent and incoherent regimes—the subject of this work. Figure 2 shows some measured |γ| dependencies on sheer Δr: Going from red to green to blue curves the optical bandwidth of the pump source increases from 0.75 to 7 to 39 nm. The central peak at Δr = 0 of all the curves has the same width indicating that the coherence radius is independent on the source bandwidth as expected. What changes with increasing bandwidth is the structure and the average level at the wings of the function |γ| outside the main peak at sheer values exceeding twice coherence radius. Since none of the curves are symmetric about zero it follows that |γ| is an inhomogeneous function, meaning that its numeric value depends on the location of the points P₁ and P₂ on the output face of the MMF, and not only on the distance between them. It is also obvious that |γ| is clearly not a 2J₁(x)/x function often assumed in such cases. Even for the largest source bandwidth (blue curve) for which case Δt/t_c = 40 ≫ 1 a large residual coherence is present at large sheer values. For Δt/t_c ∼ 1 (red and green curves) the function |γ| fluctuates strongly indicating that two arbitrary points on the MMF output face can have varying mutual coherences. The average residual coherence at large sheer values appears to be constant for a given fiber and source bandwidth and can therefore be used as a characteristic of |γ|, along with the coherence radius. In the following section we develop a simple model describing the dependence of the average residual coherence on Δt/t_c as well as on the number of modes N supported by the fiber.

 

Fig. 2 Modulus of the complex degree of coherence |γ| at the output of a step-index MMF pumped with a monochromatic source (black crosses) and a broadband source with different optical bandwidths: red triangles—0.75 nm; green circles—7 nm; blue squares—39 nm.

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3. Modeling and discussion

The experimental results described above indicate that the modulus of the complex degree of coherence |γ| is, in general, a complicated function of the transverse coordinates of the two points on the output face of the MMF, depending also parametrically on source bandwidth, modal dispersion, and as we will see later, on the number of modes. Increasing the bandwidth of the source or increasing the total modal dispersion by e.g. increasing the length of the MMF leads to lowering of the wings of this function (residual coherence) on average. The maximum and minimum values of |γ| also decrease at the wings. The full shape of the coherence function can be modeled analytically in the case of two-three modes [26]. More general case of a large, but finite number of modes can be easier approached numerically.

As our experiments suggest the structure of the function |γ| comprises the central peak for Δr within a few coherent radii and the fluctuating wings for wider separations between the two points. Our intent is to derive the statistical properties of |γ| at the wings—probability distribution and average values—as functions of two dimensionless parameters: Number of modes N and the ratio of the total modal dispersion to the coherence time of the source Δt/t_c. Using oscillatory behavior of the fiber modes in radial and azimuthal coordinates allows us to dispense with the exact expressions of the modal field distributions—a significant simplification of the analysis.

The (equal-time) degree of coherence between two points P₁ and P₂ with coordinates R₁ and R₂ is defined as [1]

Equation (1) will then contain intermodal correlators of the form $〈u_{ks}{u}_{lp}^{*}〉$ where we used a shortened notation u_ks ≡ u_k (R_s, t):

Now, assuming the spectral width of the source is not too large compared to its central frequency ω₀, the propagation constant of each mode can be expanded in a series β_k (ω) = β_k (ω₀) + τ_k (ω − ω₀), where τ_k = dβ_k/dω|_ω0 is the group delay of a given mode evaluated at the central frequency ω₀. The expression for the intermodal correlator then translates into

For the complex degree of coherence we therefore will use:

Fiber modes of not too low of an order (most of the modes in the fiber) exhibit oscillatory behavior in both radial and azimuthal coordinates. Assuming subsequent spatial averaging, we can represent the amplitudes a_ks of the mode k at a point P_s as a random number in the range from −1 to +1. More precisely, $a_{ks}=\text{sin}\left[\frac{\pi }{2}(2\xi -1)\right]$, where ξ is a random number uniformly distributed between 0 and 1. The use of sin() function makes the distribution of the amplitude a_ks slightly non-uniform, but more realistic. With this recipe the evaluation of Eq. (6) requires two columns of random modal amplitudes corresponding to two points P₁ and P₂, each column containing N random values for N modes. Multiple evaluations of absolute values of Eq. (6) are then averaged to arrive at the average residual coherence 〈|γ|〉. The two parameters of the model are the ratio of total modal dispersion to coherence time of the source Δt/t_c and the number of modes N supported by the fiber. Intuitively, the model becomes more accurate for larger N since a larger fraction of the total number of modes is high-order with rapidly oscillating spatial structure in radial and azimuthal coordinates.

Evaluation of Eq. (6) also requires the intermodal group delay values Δτ_kl. These can be derived from the modal dispersion τ_mod ≈ NA²/2nc and the number of modes N. A typical step-index multimode fiber with core radius a and numerical aperture NA supports N_total = (4aNA/λ)² modes. For example, the 105-micron core AFS105 MMF (Thorlabs) will have a total of about 800 modes at 1.55 micron wavelength. Some of these modes are degenerate, having (nearly) the same propagation constants and group delays. We have to count only non-degenerate modes in our model, however. Gloge [28] and Crosignani [12] postulate that there are $\sqrt{N_{\mathit{total}}}$ non-degenerate modes and their density on the effective index or group delay axis is non-uniform. In contrast, numerical modeling of realistic step-index MMFs indicates that degenerate modes cluster in pairs and fourths, making the total number of non-degenerate modes better estimated by N = N_total/α, where α is a number between 2 and 4. For the fiber mentioned above this is roughly 200–400 modes. We will see later that this estimate better fits the experimental results. Furthermore, modeling also shows that the group delay is a (nearly) linear function of non-degenerate mode number and mode density is nearly constant for a step-index MMF. Thus, we can write the argument of the correlation function B as Δτ_klL = Δt (k − l)/(N − 1), where k and l run from 1 to N counting all non-degenerate modes.

Figure 3 shows computed probability density plots for the values of the residual coherence |γ| as a function of Δt/t_c for both Gaussian and Lorentzian spectral shapes. The number of non-degenerate modes is used as a parameter and is shown in each panel. A vertical slice through each panel is a histogram of values that |γ| assumes for a pair of randomly picked widely separated points on the output face of the MMF. When the total modal dispersion is smaller than the coherence time of the source Δt/t_c ≪ 1 the residual coherence values are non-fluctuating and are close to 1, as expected. With increasing Δt/t_c the residual coherence begins to decrease on average and the spread of possible |γ| values increases, which is also identical to the behavior observed in our experiments. At very large values of Δt/t_c ≫ 1 all modes become incoherent (non-interfering) at which point the average residual coherence saturates at a non-zero value, which depends on the number of modes. The average value of the residual coherence 〈|γ|〉 is plotted in Fig. 4 for the two spectral shapes. The behaviors for Gaussian and Lorentzian spectra are very similar, except the average residual coherence, Fig. 4, and the minimum values from Fig. 3 are slightly smaller for Lorentzian spectrum because it has higher wings than the Gaussian. In the “saturation” regime Δt/t_c ≫ 1 any further increase in Δt/t_c yields no further reduction of average residual coherence or any change in the statistical properties of |γ|. Not surprisingly the individual curves in Fig. 4 begin to saturate when Δt/t_c ≈ N, i. e. when the group delay between any two neighboring modes is equal to the coherence time of the source. From here we can derive a simple practical condition on the fiber length required to achieve complete decoherization of the modes as $L\ge 10nc{t}_c{a}^2/{\lambda }_0^2$. The residual coherence will not be zero, however, but is determined by the number of modes in the fiber, as shown below. Furthermore, the individual values of |γ| will still fluctuate wildly for various pairs of points on the output fiber face even in the saturation regime, Fig. 3.

 

Fig. 3 Modulus of the complex degree of coherence |γ| for a pair of widely separated points at the output of a MMF on log-log scale. Left column—Gaussian spectrum, right column—Lorentzian spectrum. Number of non-degenerate modes is listed for each panel. Color represents the probability of a particular value of |γ| from 0 (deep blue) to 1 (red).

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Fig. 4 Average residual coherence from the density plots of Fig. 3 for Gaussian (left) and Lorentzian (right) spectra. Number of modes used to compute each curve is listed on the right.

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In a similar manner, the dependence of the residual coherence on the number of modes with Δt/t_c as a parameter is shown in Figs. 5 and 6. Again we observe that the shape of the source spectrum makes only a small difference with 〈|γ|〉 values slightly smaller for a high-wing Lorentzian spectrum. The dashed line in Fig. 6 corresponds to the regime of non-interfering modes and is reasonably fitted by a straight line log(〈|γ|〉) = −0.078(±0.002) − 0.5088(±0.0013)log(N) for either Gaussian or Lorentzian spectrum. This dependence is rather strikingly close to $〈\left|\gamma \right|〉~1/\sqrt{N}$ and provides a quick estimate for the residual coherence.

 

Fig. 5 Modulus of the complex degree of coherence |γ| for a pair of widely separated points at the output of a MMF as a function of the number of modes on a log-log scale. Left column—Gaussian spectrum, right column—Lorentzian spectrum. The value of the parameter Δt/t_c is listed for each panel. Color represents the probability of a particular value of |γ| from 0 (deep blue) to 1 (red).

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Fig. 6 Mean value of the complex degree of coherence |γ| from the density plots of Fig. 5 for Gaussian (left) and Lorentzian (right) spectra. Δt/t_c values are shown on the right. Dashed line corresponds to Δt/t_c = 1000.

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The predictions of this simple model were compared to the results of our experiments, described in the previous section. The data obtained using various source spectral widths is shown in Fig. 7. We compute the average residual coherence 〈|γ|〉 from multiple experimental curves similar to Fig. 2 by averaging all values of |γ| outside the main peak, defined by the first minima on both sides of zero sheer. This being not too precise of an operation it nevertheless shows a remarkable agreement between the experiment and the model. Interestingly, the residual coherence is a rather slow function of bandwidth so that even for Δt/t_c ∼ 100 the residual coherence is not fully suppressed, having a value of about 0.1. A better overlap of the data with the N = 500 dashed curve confirms that the 105-micron core step-index MMF supports several hundred non-degenerate modes, in agreement with our previous estimate. We comment, however, that at large values of Δt/t_c the modes previously thought to be (almost) degenerate may accumulate non-negligible delays among themselves thus lowering further the residual coherence level. This aspect needs to be investigated further.

 

Fig. 7 Residual coherence from experiments (circles) and the model (solid and dashed lines) as a function of Δt/t_c.

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One practical conclusion that can be derived from these results is that in order to minimize the residual coherence one must employ both a broadband source and a fiber supporting a large number of modes. Increasing only one or the other will reduce residual coherence up to a point, after which a saturation limit will be reached.

4. Conclusion

Spatial coherence properties of light at the output of a step-index multimode fiber were studied experimentally using sheering, delay-dithering Mach-Zehnder interferometer. The fiber was pumped with focused spatially-coherent optical sources of varying spectral widths from monochromatic to 40 nm wide. As the source bandwidth is increased the contribution from intermodal interference diminishes leading to the decrease of the residual coherence—average value at the wings of the modulus of the complex degree of coherence |γ| as a function of spacing Δr between the two points on the output face of the fiber. The average residual coherence is a slow decreasing function of the ratio of the fiber total modal dispersion to the source coherence time. When this ratio becomes large so that fiber modes no longer interfere the average residual coherence saturates at a non-zero level dependent on the number of modes. A simple model is developed to describe this behavior. In the regime of non-interfering modes the model predicts that the saturation level of the average residual coherence decreases roughly as inverse square-root of the number of modes. This non-interfering mode regime can be reached for sufficient fiber lengths or source bandwidths, after which no further decoherization is possible. These results and dependencies are expected to be useful in designing coherent-to-partially coherent conversion systems using multimode fibers.