Extension of light transmission distance of single-mode fiber using a microaxicon-lensed fiber end

Yuanzheng Chen; Jiwen Cui; Jiubin Tan

doi:10.1364/AO.54.006471

1. INTRODUCTION

A single-mode fiber is currently used extensively in such fields as optical communications, biomedicine, and measurement technology [1]. The light passing through a single-mode fiber has a rather large divergence angle, and the light spot does not have a clear shape. Consequently, using it to transmit a stable light signal is difficult. Furthermore, the light rapidly decays as it propagates. Thus, its effective transmission distance is limited. Much research has been conducted into ways in which the light transmission distance may be extended. For example, Kim et al. [2] proposed a three-segment fiber collimator and Lee [3] proposed a fiber lens with a polymer layer. However, the systems proposed thus far do not result in a fiber lens that is sufficiently small for application in narrow spaces. This paper investigates the efficacy of a system in which a microaxicon is at the end of the fiber to extend the light transmission distance of single-mode fiber.

2. THEORETICAL ANALYSIS

A. Principle of Bessel Beam Generated by Microaxicon

A diffraction free light beam (Bessel beam) was discussed by Durnin et al. [4]. It is a particular solution to the wave equation in which most of the light energy is focused on the principal axis, resulting in its spot having a sharp clearance. A true Bessel beam would never diffract; hence, the intensity on the principal axis is irrelative to the propagation distance. Consequently, the energy possessed by a Bessel beam as it propagates over a long distance is high [5].

A true Bessel beam cannot be generated because it requires infinite energy; however, an approximation can be generated using a Gaussian beam and a microaxicon [6]. As shown in Fig. 1, interference can be produced by passing a light beam through a microaxicon. The light beam in the interference field is then an approximation of the Bessel beam.

Fig. 1. Bessel beams generated using a microaxicon.

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In Fig. 1, $λ$ is the wavelength of the incident wave, $n$ is the refractive index of the microaxicon, $z$ is the propagation distance, $k$ is the angular wavenumber, and $φ$ is the base angle of the microaxicon. The transmission function of the microaxicon can thus be expressed as

t (r_{1}) = {\begin{cases} \exp [- i k (n - 1) φ r_{1}], & r_{1} \leq R \\ 0 & r_{1} > R \end{cases},

where

r_{1}

and

r_{2}

are the radial coordinates on the incident and observation planes, respectively. The incident wave equation can be expressed as

E_{i} (r_{2}, z) = \exp (\frac{- r_{2}^{2}}{ω^{2}}),

where

ω

is the radius of the Gaussian beam. Using Collins’ formula, the wave equation of light can be calculated as

E_{o} (r_{2}, z) = - \frac{i k}{z} \exp (\frac{i k}{2 z}) \int_{0}^{\infty} [\exp (- \frac{r_{2}^{2}}{ω^{2}}) t (r_{2}) \times \exp (\frac{i k r_{2}^{2}}{2 z}) J_{0} (\frac{k r_{1} r_{2}}{z}) r_{2} d r_{2}] .

Further, by substituting Eq. (1) into Eq. (3), we obtain

E_{o} (r_{2}, z) = - \frac{i k}{z} \exp (\frac{i k}{2 z}) \int_{0}^{R} {\exp [i k (\frac{r_{2}^{2}}{2 z} - (n - 1) φ r_{2})] \times \exp (\frac{- r_{2}^{2}}{ω^{2}}) J_{0} (\frac{k r_{1} r_{2}}{z}) r_{2} d r_{2}} .

We can now define the phase function as

h (r_{2}) = \frac{r_{2}^{2}}{2 z} - (n - 1) φ r_{2} .

The maximum light transmission distance of the microaxicon designed can be expressed as

\frac{d h (r_{2})}{d r_{2}} = \frac{r_{2}}{z} - (n - 1) φ = 0,

where

z_{\max} = \frac{R}{(n - 1) φ} .

When

z < Z_{\max}

, the light intensity on the principal axis is given by

I (0, z) \approx α^{2} λ z J_{0}^{2} (α r_{2}),

where

α = k (n - 1) φ

.

As described by Eq. (8), the light intensity increases linearly with light propagation on the principal axis.

B. Characteristics of Bessel Beams Generated by Microaxicon

To investigate the distribution of light in the far field, a simulation was performed (Fig. 2).

Fig. 2. Simulation arrangement.

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In the simulation, prior to the interference being formed, the light spot was a ring, as shown in Fig. 3. However, because of the light beam interference, as it propagated further, it became a series of concentric circles. Further propagation then resulted in the light beam being focused on the principal axis.

Fig. 3. Spot images at different positions. (a) Ring. (b) Concentric circles. (c) Dot.

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The effectiveness of the microaxicon fixed at the end of the fiber was also investigated via simulation (Fig. 4). In the simulation, an interference field was created 20 μm away from the fiber end with the microaxicon at a base angle of 45°. A light beam passed through this field and was enhanced as the propagation distance increased. The interference reached its maximum intensity at 80 μm away from the end of the fiber end and then decayed rapidly as the light propagated further. Compared with a flat fiber end, the light focused much more energy on the principal axis using the microaxicon fabricated at the fiber end. Light emitting from the microaxicon remained at a high intensity on the principal axis at a distance of 20–80 μm away from the fiber end.

Fig. 4. Light simulations (a) passing through a microaxicon with a base angle of 45° and (b) passing through a flat fiber end.

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As shown in Fig. 5, when the radius of microaxicon $R$ fell in the range 20–50 μm, with the base angle constant, the maximum collimation distance increased as the radius of the microaxicon increased. The maximum collimation distance decreased as the base angle of the microaxicon increased when the radius remained constant.

Fig. 5. Relation among radius of microaxicon, base angle, and maximum collimation distance.

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As shown in Fig. 6, when the base angle of the microaxicon fixed at the fiber end increased, the interference field moved toward the fiber end. Further, the length of the interference field decreased as the base angle of the microaxicon increased. The simulation results indicate that a maximum transmission distance of 80 μm is achievable with the microaxicon at a base angle of 45°.

Fig. 6. Light intensity distribution on the principal axis.

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The results of investigations into the intensity distribution of the light passing through a 45° microaxicon via simulation are shown in Fig. 7. As can be seen, the light spot had a clear shape in the range 70–80 μm away from the fiber end. The high light intensity at a distance of 80 μm away from the fiber end indicates that this light beam is a long-distance transfer light beam. In the figure, $h$ is the distance between the incident and chief rays; the simulation was conducted with $h$ in the range 0–30 μm.

Fig. 7. Intensity distribution of light passing through a 45° microaxicon.

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3. EXPERIMENTAL RESULTS

A. Stability of Light

A microaxicon with a base angle of 45° was fabricated at the fiber end by polishing the fiber end with a fiber polisher, as shown in Fig. 8.

Fig. 8. Microaxicon lensed fiber used in the experiment.

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The stability of the light passing through the microaxicon at the fiber end was verified via an experimental study, as illustrated in Fig. 9.

Fig. 9. Experimental environment used to verify the stability of the light passing through the microaxicon.

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$C P_{\max}$ and $C P_{\min}$ are the respective maximum and minimum values of the centroid position of the image. The single-pixel size of the CCD used in our experiment was 4.4 μm. The image stability could be calculated as

P V = C P_{\max} - C P_{\min} .

As shown in Fig. 10, an image stability of 1.32 μm was achieved with the microaxicon fixed at the fiber compared with 1.72 μm at the flat end. Thus, the microaxicon at the fiber end improved the image stability at the fiber end by 23%.

Fig. 10. Stability of spot image (a) at flat fiber end (b) at microaxicon lensed fiber end.

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Using a variable $R_{pds}$ to signify the stability of the light beam in the far field, $D$ as the propagation distance, $S_{\max}$ as the maximum value of the shift, and $S_{\min}$ as the minimum value of the shift, we obtain Eq. (10):

R_{pds} = \frac{(S_{\max} - S_{\min})}{D} .

The experimental results indicate that an $R_{pds}$ of 8.97 pixel/mm was achieved in the far field with the flat fiber end compared with an $R_{pds}$ of 0.2 pixel/mm using the microaxicon at the fiber end. As shown in Fig. 11, the stability of the light beam improved by 97% with the microaxicon at the fiber end. This indicates that, by generating Bessel beams, the microaxicon at the fiber end focused the light beam and reduced the noise inside the fiber, thereby improving the stability of the light beam. A light beam stability of 0.2 pixel/mm was achieved 80 μm away from the microaxicon-lensed fiber end. As shown in Fig. 7, this experimental result is also supported by the simulation results. The light stability achieved its maximum value because the light intensity on the principal axis achieved a maximum value. Thus, it can be concluded that the microaxicon at the fiber end made the light beam an effective long-distance light signal.

Fig. 11. Stability of light beam passing through the flat fiber end and the microaxicon-lensed fiber end in far field.

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B. Light Collimation Experiment

The efficacy of light collimation with the microaxicon fixed at the fiber end was also verified experimentally, as shown in Fig. 12. The divergence of light was calculated using the following equation:

\tan θ = \frac{R - r}{s},

where

R

is the radius of the light spot,

s

is the distance between the fiber end and the image plane, and

r

is the radius of the fiber end.

Fig. 12. Calculation of divergence angle.

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The experimental results indicate that the divergence angle $θ$ of the light beam passing through the flat fiber end was reduced from 4.1° to 0.47° with the microaxicon at the fiber end, as shown in Fig. 13. It therefore can be concluded that the microaxicon at the fiber end reduced the divergence angle of the light beam at 80 μm from the fiber end, making the light beam an effective long-distance transfer beam.

Fig. 13. Relation between propagation distance and spot radius.

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C. Light Distribution in Far Field Experiment

The arrangement of the sampling points shown in Fig. 14 can be used to depict the light distribution in the entire far field because of the axial symmetry of light distribution.

Fig. 14. Arrangement of sampling points.

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As shown in Fig. 15, the light distribution was more uniform in the range 30–50 μm away from the fiber end. Further, the light spot had a clear shape in the range 70–80 μm away from the fiber end. Thus, a uniform light and a clear light spot were simultaneously achieved. If a uniform light spot is needed, light in the range 30–50 μm away from the fiber end can be used. Conversely, if a clear light spot is needed, light in the range 70–80 μm away from the fiber end can be used. Compared with light distribution in the far field without the microaxicon lens, the light passing through the microaxicon retained a high intensity at 80 μm away from the fiber end. It therefore can be concluded that the microaxicon at the fiber end enabled the light beam to be used as a long-distance transfer light beam.

Fig. 15. Distribution of light in far field: (a) with microaxicon-lensed fiber; (b) without the microaxicon lens.

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4. CONCLUSION

In this study, a microaxicon was designed at the end of single-mode fiber to extend its light transmission distance. The results of analysis of the characteristics of the light emitting from the microaxicon via simulations indicate that a maximum transmission distance of 80 μm was achieved with the microaxicon at a base angle of 45°. The effectiveness of the microaxicon was also verified experimentally, with the results indicating that, compared with a flat end, the divergence angle of light was reduced from 4.1° to 0.47°, and an increase of 80 μm in the light transmission distance of single-mode fiber was achieved. Compared with a flat end fiber, the stability of light was improved by 97% when the microaxicon was used. Further, the light spot had a sharp clearance in the range 70–80 μm away from the fiber end. Thus, this light can be utilized in various ways to meet a variety of requirements. Furthermore, the high light intensity at a distance of 80 μm away from the fiber end signifies that this light beam can be used as a long-distance transfer light beam. It therefore can be concluded that a microaxicon at the fiber end effectively extends the light transmission distance of single-mode fiber.

Funding

National Natural Science Foundation of China (NSFC) (51175128).

REFERENCES

1. J. Cui, Y. Chen, and J. Tan, “Improvement of dimensional measurement accuracy of microstructures with high aspect ratio with a spherical coupling fiber probe,” Meas. Sci. Technol. 25, 075902 (2014). [CrossRef]

2. J. Kim, M. Han, S. Chang, J. W. Lee, and K. Oh, “Achievement of large spot size and long collimation length using UV curable self-assembled polymer lens on a beam expanding core-less silica fiber,” IEEE Photon. Technol. Lett. 16, 2499–2501 (2004). [CrossRef]

3. J. Lee, “Collimation scheme for fiber lens using polymer coating,” in International Conference on Computing and Communications Strategies (IEEE, 2012), pp. 1423–1425.

4. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef]

5. R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, “Imaging properties of axicon in a scanning optical system,” Appl. Opt. 31, 6653–6657 (1992). [CrossRef]

6. M. Lei and B. Yao, “Characteristics of beam profile of Gaussian beam passing through an axicon,” Opt. Commun. 239, 367–372 (2004). [CrossRef]

Extension of light transmission distance of single-mode fiber using a microaxicon-lensed fiber end

Abstract

1. INTRODUCTION

2. THEORETICAL ANALYSIS

A. Principle of Bessel Beam Generated by Microaxicon

B. Characteristics of Bessel Beams Generated by Microaxicon

3. EXPERIMENTAL RESULTS

A. Stability of Light

B. Light Collimation Experiment

C. Light Distribution in Far Field Experiment

4. CONCLUSION

Funding

REFERENCES

Cited By

Figures (15)

Equations (11)

Applied Optics