1. Introduction

As we launch the laser light into a step-index multimode optical fiber, the optical pattern at the fiber output endface is found to be split into many irregular light spots, which are called fiber speckles [1, 2]. They result from the fact that the laser light is of high optical coherence. The field interference of higher-order modes produces the observed optical patterns.

The method of fiber mode scrambling has been proposed to obtain a uniformly distributed output light beam emanating from a multimode fiber [3]. Here, we report another easier method by reforming the input laser light into a partially incoherent light source to enhance the output beam uniformity as well as improve the fiber cross-sectional imaging [1]. This approach can also improve the fiber index profiling [4], and provides a much more practical solution for various applications where multimode fibers are used as the transmission media, such as in medical imaging, disease diagnosis, and fiber based optical trapping [5–7]. To explore the fiber coupling characteristics, various mode patterns of a multimode optical fiber are also calculated theoretically and demonstrated experimentally.

2. Generation of low coherence light

Figure 1 shows the setup used to transform the incident laser light into a low coherence light source and demonstrates how to control the optical coherence of the output light. We make use of a pair of lenses (focal length f ₁=50 mm and f ₂=150 mm) to expand and collimate the He-Ne laser beam (wavelength λ=632.8 nm), and let the beam pass through another focusing lens (f ₃=50 mm). Near the laser focus, we put a semi-transparent diffuser which can be driven by a fast rotating motor. After transmitting the diffuser, the laser beam is scattered into many tiny light spots, which behave as many new point light sources.

 

Fig. 1. The experimental setup for transforming a coherent laser light into a low coherence light and observing the optical speckles.

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We then adjust the relative positions of the diffuser away from the laser focus and observe the far-field optical speckles of the transmitted light which are projected on a screen. We have observed that as the distance between the diffuser and the laser focus grows larger, the width of the laser beam hit on the diffuser becomes bigger and much more tiny light spots are scattered, reducing the optical coherence to a lower degree [8].

As the diffuser is rotating, all the light spots move and change randomly such that the averaged light intensity appears to be smoothly distributed. Nevertheless, the optical coherence decreases as a consequence of the instantaneous fragmented beam shape [9, 10]. This low coherence light source will be shown to be capable of improving the cross-sectional imaging of optical fibers and enhancing the output beam uniformity.

3. Application in the fiber cross-sectional imaging

Furthermore, as shown in Fig. 2, we couple the reformed light beam of which the coherence level can be manually controlled into a step-index multimode optical fiber (AFS50/125Y, THORLABS Inc., length ≈6 cm) by two microscope objective lenses (10X and 20X), and project the fiber output image onto a CCD camera by another microscope objective lens (10X). We would like to explore how the output cross-sectional images are related to the degree of optical coherence in the cases of a stopped diffuser (Fig. 3) and a rotating diffuser (Fig. 4), respectively.

 

Fig. 2. The experimental setup for coupling a low coherence light into a test fiber. MO, microscope objective lens; PC, personal computer; CCD, charge coupled device.

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As shown in Figs. 3 and 4, when the diffuser is moved far from the laser focus, the observed fiber image reveals that the fiber core turns to be smoother due to the rotation of the diffuser and the fiber cladding becomes brighter due to the diffraction of the tiny light spots inside the optical fiber. The smaller the size of the light spot scattered by the diffuser, the larger is the diffraction angle of the split tiny light beam, and the higher is the mode order of the light coupled into the optical fiber. We can observe that more light can penetrate into the fiber cladding when the optical coherence of the input light is reduced lower, and we can thus obtain a more clear and uniform fiber cross-sectional image. If the degree of optical coherence could be made much lower by further moving the rotating diffuser away from the laser focus, we should be able to capture the fiber image of much higher brightness and uniformity.

 

Fig. 3. The variation of observed fiber speckles with the distance between the diffuser and the laser focus when the diffuser is stopped.

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Fig. 4. The variation of observed fiber images with the distance between the diffuser and the laser focus when the diffuser is rotating.

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4. Simulation and observation of the fiber mode patterns

The improvement in the fiber cross-sectional imaging by using the low coherence light is attributed to the enhanced light coupling of the increased higher-order modes from the fiber core into the fiber cladding. As a consequence, we intend to investigate the mode patterns inside a multimode optical fiber numerically and experimentally [11, 12].

The simulation is performed using the radial effective-index method [13]. Inside the fiber, the propagation constant β along the fiber central axis and the wave number k of the light beam have the relationship β=k cosθ, where θ is the angle between them. We calculate the mode patterns by gradually reducing the effective refractive index n_eff=β/k ₀, where k ₀ is the free space wave number. The action corresponds to increasing the incident angle α or the refraction angle θ of the light beam at the fiber input face. We assume the fiber is lossless.

The step-index multimode fiber used in the simulation has a core diameter of 50 µm, a cladding diameter of 125 µm, a core index n ₁=1.4457, and a cladding index n ₂=1.4378. The fiber’s numerical aperture $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ is 0.151 and the maximum incident angle α _max=sin^-1NA is 8.68° for a guided mode in the fiber core. The optical wavelength is assumed to be 632.8 nm. The values of n ₁ and n ₂ are adopted from the common prescribed values used in the simulation software. Though the nominal NA is 0.22 at 1060 nm wavelength according to the THORLABS catalog, we continue to utilize the index values aforementioned in the simulation due to the agreement between the mode calculation and measurement.

The simulation results of the optical patterns of several representative modes inside the multimode fiber are depicted in Fig. 5. The core modes [Figs. 5(a)–(h)] can be observed as the effective refractive index has a relation of n ₁>n_eff>n ₂. The cladding modes [Figs. 5(i)–(l)] can be observed as the n_eff value is reduced across the cladding index n ₂. Figure 5(g) shows a donut-shaped core mode or a hollow beam [14, 15], and Fig. 5(j) shows a donut-shaped cladding mode. The higher-order cladding modes finally will evolve into many tiny light spots which are uniformly distributed in the whole fiber as shown in Figs. 5(k) and 5(l). The spot size will shrink with the decreased n_eff value (or the increased incident angle α).

 

Fig. 5. Calculated mode patterns of a multimode fiber as the n_eff value is set at (a) 1.4456, (b) 1.44557, (c) 1.44555, (d) 1.4454, (e) 1.4453, (f) 1.4450, (g) 1.4410, (h) 1.4380, (i) 1.4370, (j) 1.4360, (k) 1.4340, (l) 1.4250, respectively.

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The experimental setup is shown in Fig. 6. We focus the He-Ne laser light beam by a 20X microscope objective lens into a test fiber, which is mounted on a rotation stage and a three-dimensional translation stage. The mode patterns on the output face of the test fiber are projected onto a CCD camera through another 10X microscope objective lens which acts as an imaging lens.

 

Fig. 6. The experimental setup for coupling a coherent laser light into a test fiber under various incident angles.

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We take a step-index multimode fiber (AFS50/125Y, THORLABS Inc., length ≈6 cm) to test the mode patterns by coupling the He-Ne laser light at various incident angles. The experimental results of the observed fiber cross-sectional images using a coherent laser light are shown in Fig. 7. Figures 7(a) and 7(b) show the guided modes in the fiber core for the incident angles α at 1° and 1.5°. We can find that the size of the massive light spots in the core mode will be reduced as we increase the incident angle, coinciding with the trend observed in the simulation results [Figs. 5(a)–(h)]. Figure 7(c) for α=4° shows a donut-shaped cladding mode mixed with some residual core mode, which corresponds to the calculated mode pattern in Fig. 5(j). Figure 7(d) for α=15° exhibits highly fragmented optical speckles which are uniformly distributed in the whole fiber and correspond to the calculated mode patterns in Figs. 5(k) and 5(l). Here, we can also find that the spot size will further shrink with the increased incident angle α (or the decreased n_eff value).

On the other hand, the incident angle α=4° of the launched laser light with regard to Fig. 7(c) is smaller than the maximum incident angle α _max for a core mode, so the dominant donut-shaped cladding mode could be observed only when the majority of the laser light is coupled into the fiber at the cladding region, having $n_{\mathrm{eff}}=n_2\cos \theta =\sqrt{n_2^2-{\left(\sin \alpha \right)}^2}=1.4361$ which is very close to the n_eff value of the calculated mode pattern in Fig. 5(j). The weak edge of the launched laser light happens to be coupled into the fiber at the core region, since the $n_{\mathrm{eff}}=n_1\cos \theta =\sqrt{n_1^2-{\left(\sin \alpha \right)}^2}=1.4440$ is bigger than the cladding index n ₂, forming a guided mode in the core. Hence, Fig. 7(c) reveals that multiple modes are excited. Figure 7(d) for α=15° reveals that the laser light coupled into the core has n_eff=1.4223, which is smaller than the cladding index n ₂, such that it will propagate in both the core and the cladding.

The weak rim of the focused low coherence light (Fig. 2) which has a larger beam width than the coherent laser light can be coupled into the fiber directly at the cladding region as indicated by Fig. 7(c), and the increased tiny light spots of large enough diffraction angles in the core can penetrate into the cladding as indicated by Fig. 7(d). All the above results have clearly elucidated why the fiber cladding will become brighter (Figs. 3 and 4) when the coherence level of the input light is made lower.

 

Fig. 7. Observed cross-sectional images of a multimode fiber for the incident angles α of the coherent laser light at (a) 1°, (b) 1.5°, (c) 4°, (d) 15°, respectively.

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5. Conclusion

When we pass the focused laser beam through a variable-distance rotating diffuser, we can control the coherence level of the transmitted light as well as the spot size of the optical speckles. If we couple this manually controlled low coherence light into a multimode optical fiber, we can obtain uniform fiber cross-sectional images of higher quality. The mechanism is explained by the numerical simulation and the experimental demonstration of the mode patterns inside a multimode optical fiber. This approach of improving the output beam uniformity is an excellent and efficient solution especially for various medical applications where multimode fibers are required.